Господин Экзамен
Упрощение выражений/
2563*(sin(2*x)/2+x)/800-35*(x-sin(2*x)/2)/2-2563*sin(x)/200+211*cos(x)^2/25-422*cos(x)/25+2563*x/400 если x=2
2563*(sin(2*x)/2+x)/800-35*(x-sin(2*x)/2)/2-2563*sin(x)/200+211*cos(x)^2/25-422*cos(x)/25+2563*x/400 если x=2
Решение
Вы ввели
[src]
/sin(2*x) \ / sin(2*x)\
2563*|-------- + x| 35*|x - --------| 2
\ 2 / \ 2 / 2563*sin(x) 211*cos (x) 422*cos(x) 2563*x
------------------- - ----------------- - ----------- + ----------- - ---------- + ------
800 2 200 25 25 400
$$\frac{211 \cos^{2}{\left(x \right)}}{25} + \frac{2563 x}{400} - \frac{35 \left(x - \frac{\sin{\left(2 x \right)}}{2}\right)}{2} + \frac{2563 \left(x + \frac{\sin{\left(2 x \right)}}{2}\right)}{800} - \frac{2563 \sin{\left(x \right)}}{200} - \frac{422 \cos{\left(x \right)}}{25}$$
2563*(sin(2*x)/2 + x)/800 - 35*(x - sin(2*x)/2)/2 - 2563*sin(x)/200 + 211*cos(x)^2/25 - 422*cos(x)/25 + 2563*x/400
Общее упрощение
[src]
2 6311*x 2563*sin(x) 422*cos(x) 211*cos (x) 16563*sin(2*x) - ------ - ----------- - ---------- + ----------- + -------------- 800 200 25 25 1600
$$\frac{211 \cos^{2}{\left(x \right)}}{25} - \frac{6311 x}{800} - \frac{2563 \sin{\left(x \right)}}{200} + \frac{16563 \sin{\left(2 x \right)}}{1600} - \frac{422 \cos{\left(x \right)}}{25}$$
-6311*x/800 - 2563*sin(x)/200 - 422*cos(x)/25 + 211*cos(x)^2/25 + 16563*sin(2*x)/1600
Подстановка условия
[src]
2563*(sin(2*x)/2 + x)/800 - 35*(x - sin(2*x)/2)/2 - 2563*sin(x)/200 + 211*cos(x)^2/25 - 422*cos(x)/25 + 2563*x/400 при x = 2
подставляем
/sin(2*x) \ / sin(2*x)\
2563*|-------- + x| 35*|x - --------| 2
\ 2 / \ 2 / 2563*sin(x) 211*cos (x) 422*cos(x) 2563*x
------------------- - ----------------- - ----------- + ----------- - ---------- + ------
800 2 200 25 25 400
$$\frac{211 \cos^{2}{\left(x \right)}}{25} + \frac{2563 x}{400} - \frac{35 \left(x - \frac{\sin{\left(2 x \right)}}{2}\right)}{2} + \frac{2563 \left(x + \frac{\sin{\left(2 x \right)}}{2}\right)}{800} - \frac{2563 \sin{\left(x \right)}}{200} - \frac{422 \cos{\left(x \right)}}{25}$$
2 6311*x 2563*sin(x) 422*cos(x) 211*cos (x) 16563*sin(2*x) - ------ - ----------- - ---------- + ----------- + -------------- 800 200 25 25 1600
$$\frac{211 \cos^{2}{\left(x \right)}}{25} - \frac{6311 x}{800} - \frac{2563 \sin{\left(x \right)}}{200} + \frac{16563 \sin{\left(2 x \right)}}{1600} - \frac{422 \cos{\left(x \right)}}{25}$$
переменные
x = 2
$$x = 2$$
2
6311*(2) 2563*sin((2)) 422*cos((2)) 211*cos ((2)) 16563*sin(2*(2))
- -------- - ------------- - ------------ + ------------- + ----------------
800 200 25 25 1600
$$\frac{211 \cos^{2}{\left((2) \right)}}{25} - \frac{6311 (2)}{800} - \frac{2563 \sin{\left((2) \right)}}{200} + \frac{16563 \sin{\left(2 (2) \right)}}{1600} - \frac{422 \cos{\left((2) \right)}}{25}$$
2 6311 2563*sin(2) 422*cos(2) 211*cos (2) 16563*sin(2*2) - ----*2 - ----------- - ---------- + ----------- + -------------- 800 200 25 25 1600
$$\left(- \frac{6311}{800}\right) 2 - \frac{2563 \sin{\left(2 \right)}}{200} + \frac{16563 \sin{\left(2 \cdot 2 \right)}}{1600} + \frac{211 \cos^{2}{\left(2 \right)}}{25} - \frac{422 \cos{\left(2 \right)}}{25}$$
2 6311 2563*sin(2) 422*cos(2) 211*cos (2) 16563*sin(4) - ---- - ----------- - ---------- + ----------- + ------------ 400 200 25 25 1600
$$- \frac{6311}{400} - \frac{2563 \sin{\left(2 \right)}}{200} + \frac{16563 \sin{\left(4 \right)}}{1600} + \frac{211 \cos^{2}{\left(2 \right)}}{25} - \frac{422 \cos{\left(2 \right)}}{25}$$
-6311/400 - 2563*sin(2)/200 - 422*cos(2)/25 + 211*cos(2)^2/25 + 16563*sin(4)/1600
Собрать выражение
[src]
2 6311*x 2563*sin(x) 422*cos(x) 211*cos (x) 16563*sin(2*x) - ------ - ----------- - ---------- + ----------- + -------------- 800 200 25 25 1600
$$\frac{211 \cos^{2}{\left(x \right)}}{25} - \frac{6311 x}{800} - \frac{2563 \sin{\left(x \right)}}{200} + \frac{16563 \sin{\left(2 x \right)}}{1600} - \frac{422 \cos{\left(x \right)}}{25}$$
211 6311*x 2563*sin(x) 422*cos(x) 211*cos(2*x) 16563*sin(2*x) --- - ------ - ----------- - ---------- + ------------ + -------------- 50 800 200 25 50 1600
$$- \frac{6311 x}{800} - \frac{2563 \sin{\left(x \right)}}{200} + \frac{16563 \sin{\left(2 x \right)}}{1600} - \frac{422 \cos{\left(x \right)}}{25} + \frac{211 \cos{\left(2 x \right)}}{50} + \frac{211}{50}$$
211/50 - 6311*x/800 - 2563*sin(x)/200 - 422*cos(x)/25 + 211*cos(2*x)/50 + 16563*sin(2*x)/1600
Объединение рациональных выражений
[src]
2
-27008*cos(x) - 20504*sin(x) - 12622*x + 13504*cos (x) + 16563*sin(2*x)
-----------------------------------------------------------------------
1600
$$\frac{13504 \cos^{2}{\left(x \right)} - 12622 x - 20504 \sin{\left(x \right)} + 16563 \sin{\left(2 x \right)} - 27008 \cos{\left(x \right)}}{1600}$$
(-27008*cos(x) - 20504*sin(x) - 12622*x + 13504*cos(x)^2 + 16563*sin(2*x))/1600
Комбинаторика
[src]
2 6311*x 2563*sin(x) 422*cos(x) 211*cos (x) 16563*sin(2*x) - ------ - ----------- - ---------- + ----------- + -------------- 800 200 25 25 1600
$$\frac{211 \cos^{2}{\left(x \right)}}{25} - \frac{6311 x}{800} - \frac{2563 \sin{\left(x \right)}}{200} + \frac{16563 \sin{\left(2 x \right)}}{1600} - \frac{422 \cos{\left(x \right)}}{25}$$
-6311*x/800 - 2563*sin(x)/200 - 422*cos(x)/25 + 211*cos(x)^2/25 + 16563*sin(2*x)/1600
Общий знаменатель
[src]
2 6311*x 2563*sin(x) 422*cos(x) 211*cos (x) 16563*sin(2*x) - ------ - ----------- - ---------- + ----------- + -------------- 800 200 25 25 1600
$$\frac{211 \cos^{2}{\left(x \right)}}{25} - \frac{6311 x}{800} - \frac{2563 \sin{\left(x \right)}}{200} + \frac{16563 \sin{\left(2 x \right)}}{1600} - \frac{422 \cos{\left(x \right)}}{25}$$
-6311*x/800 - 2563*sin(x)/200 - 422*cos(x)/25 + 211*cos(x)^2/25 + 16563*sin(2*x)/1600
Тригонометрическая часть
[src]
2 6311*x 2563*sin(x) 422*cos(x) 211*cos (x) 16563*sin(2*x) - ------ - ----------- - ---------- + ----------- + -------------- 800 200 25 25 1600
$$\frac{211 \cos^{2}{\left(x \right)}}{25} - \frac{6311 x}{800} - \frac{2563 \sin{\left(x \right)}}{200} + \frac{16563 \sin{\left(2 x \right)}}{1600} - \frac{422 \cos{\left(x \right)}}{25}$$
6311*x 2563 422 211 16563
- ------ - ---------- - --------- + ---------- + -------------
800 200*csc(x) 25*sec(x) 2 1600*csc(2*x)
25*sec (x)
$$- \frac{6311 x}{800} - \frac{422}{25 \sec{\left(x \right)}} + \frac{16563}{1600 \csc{\left(2 x \right)}} - \frac{2563}{200 \csc{\left(x \right)}} + \frac{211}{25 \sec^{2}{\left(x \right)}}$$
211 6311*x 2563*sin(x) 422*cos(x) 211*cos(2*x) 16563*sin(2*x) --- - ------ - ----------- - ---------- + ------------ + -------------- 50 800 200 25 50 1600
$$- \frac{6311 x}{800} - \frac{2563 \sin{\left(x \right)}}{200} + \frac{16563 \sin{\left(2 x \right)}}{1600} - \frac{422 \cos{\left(x \right)}}{25} + \frac{211 \cos{\left(2 x \right)}}{50} + \frac{211}{50}$$
/ pi\ / pi\
2563*cos|x - --| 2 16563*cos|2*x - --|
6311*x \ 2 / 422*cos(x) 211*cos (x) \ 2 /
- ------ - ---------------- - ---------- + ----------- + -------------------
800 200 25 25 1600
$$\frac{211 \cos^{2}{\left(x \right)}}{25} - \frac{6311 x}{800} - \frac{422 \cos{\left(x \right)}}{25} - \frac{2563 \cos{\left(x - \frac{\pi}{2} \right)}}{200} + \frac{16563 \cos{\left(2 x - \frac{\pi}{2} \right)}}{1600}$$
/ pi\ 2/ pi\
422*sin|x + --| 211*sin |x + --|
6311*x 2563*sin(x) \ 2 / \ 2 / 16563*sin(2*x)
- ------ - ----------- - --------------- + ---------------- + --------------
800 200 25 25 1600
$$\frac{211 \sin^{2}{\left(x + \frac{\pi}{2} \right)}}{25} - \frac{6311 x}{800} - \frac{2563 \sin{\left(x \right)}}{200} + \frac{16563 \sin{\left(2 x \right)}}{1600} - \frac{422 \sin{\left(x + \frac{\pi}{2} \right)}}{25}$$
6311*x 2563 422 211 16563
- ------ - --------------- - --------- + ---------- + ------------------
800 / pi\ 25*sec(x) 2 / pi\
200*sec|x - --| 25*sec (x) 1600*sec|2*x - --|
\ 2 / \ 2 /
$$- \frac{6311 x}{800} + \frac{16563}{1600 \sec{\left(2 x - \frac{\pi}{2} \right)}} - \frac{2563}{200 \sec{\left(x - \frac{\pi}{2} \right)}} - \frac{422}{25 \sec{\left(x \right)}} + \frac{211}{25 \sec^{2}{\left(x \right)}}$$
6311*x 2563 422 211 16563
- ------ - ---------- - -------------- + --------------- + -------------
800 200*csc(x) /pi \ 2/pi \ 1600*csc(2*x)
25*csc|-- - x| 25*csc |-- - x|
\2 / \2 /
$$- \frac{6311 x}{800} - \frac{422}{25 \csc{\left(- x + \frac{\pi}{2} \right)}} + \frac{16563}{1600 \csc{\left(2 x \right)}} - \frac{2563}{200 \csc{\left(x \right)}} + \frac{211}{25 \csc^{2}{\left(- x + \frac{\pi}{2} \right)}}$$
6311*x 2563 422 211 16563
- ------ - --------------- - --------- + ---------- + ------------------
800 /pi \ 25*sec(x) 2 /pi \
200*sec|-- - x| 25*sec (x) 1600*sec|-- - 2*x|
\2 / \2 /
$$- \frac{6311 x}{800} - \frac{2563}{200 \sec{\left(- x + \frac{\pi}{2} \right)}} + \frac{16563}{1600 \sec{\left(- 2 x + \frac{\pi}{2} \right)}} - \frac{422}{25 \sec{\left(x \right)}} + \frac{211}{25 \sec^{2}{\left(x \right)}}$$
/ pi\ /pi \
422*sin|x + --| 211*sin|-- + 2*x|
211 6311*x 2563*sin(x) \ 2 / \2 / 16563*sin(2*x)
--- - ------ - ----------- - --------------- + ----------------- + --------------
50 800 200 25 50 1600
$$- \frac{6311 x}{800} - \frac{2563 \sin{\left(x \right)}}{200} + \frac{16563 \sin{\left(2 x \right)}}{1600} - \frac{422 \sin{\left(x + \frac{\pi}{2} \right)}}{25} + \frac{211 \sin{\left(2 x + \frac{\pi}{2} \right)}}{50} + \frac{211}{50}$$
211 6311*x 2563 422 211 16563
--- - ------ - --------------- - --------- + ----------- + ------------------
50 800 / pi\ 25*sec(x) 50*sec(2*x) / pi\
200*sec|x - --| 1600*sec|2*x - --|
\ 2 / \ 2 /
$$- \frac{6311 x}{800} + \frac{211}{50} + \frac{16563}{1600 \sec{\left(2 x - \frac{\pi}{2} \right)}} - \frac{2563}{200 \sec{\left(x - \frac{\pi}{2} \right)}} + \frac{211}{50 \sec{\left(2 x \right)}} - \frac{422}{25 \sec{\left(x \right)}}$$
/ pi\ / pi\
2563*cos|x - --| 16563*cos|2*x - --|
211 6311*x \ 2 / 422*cos(x) 211*cos(2*x) \ 2 /
--- - ------ - ---------------- - ---------- + ------------ + -------------------
50 800 200 25 50 1600
$$- \frac{6311 x}{800} - \frac{422 \cos{\left(x \right)}}{25} + \frac{211 \cos{\left(2 x \right)}}{50} - \frac{2563 \cos{\left(x - \frac{\pi}{2} \right)}}{200} + \frac{16563 \cos{\left(2 x - \frac{\pi}{2} \right)}}{1600} + \frac{211}{50}$$
/x\
2 2563*(1 + cos(x))*tan|-|
6311*x 422*cos(x) 211*cos (x) 16563*sin(2*x) \2/
- ------ - ---------- + ----------- + -------------- - ------------------------
800 25 25 1600 200
$$- \frac{2563 \left(\cos{\left(x \right)} + 1\right) \tan{\left(\frac{x}{2} \right)}}{200} + \frac{211 \cos^{2}{\left(x \right)}}{25} - \frac{6311 x}{800} + \frac{16563 \sin{\left(2 x \right)}}{1600} - \frac{422 \cos{\left(x \right)}}{25}$$
211 6311*x 2563 422 211 16563
--- - ------ - ---------- - -------------- + ---------------- + -------------
50 800 200*csc(x) /pi \ /pi \ 1600*csc(2*x)
25*csc|-- - x| 50*csc|-- - 2*x|
\2 / \2 /
$$- \frac{6311 x}{800} + \frac{211}{50} - \frac{422}{25 \csc{\left(- x + \frac{\pi}{2} \right)}} + \frac{211}{50 \csc{\left(- 2 x + \frac{\pi}{2} \right)}} + \frac{16563}{1600 \csc{\left(2 x \right)}} - \frac{2563}{200 \csc{\left(x \right)}}$$
6311*x 2563 422 211 16563
- ------ - --------------- - -------------- + --------------- + ------------------
800 200*csc(pi - x) /pi \ 2/pi \ 1600*csc(pi - 2*x)
25*csc|-- - x| 25*csc |-- - x|
\2 / \2 /
$$- \frac{6311 x}{800} - \frac{422}{25 \csc{\left(- x + \frac{\pi}{2} \right)}} - \frac{2563}{200 \csc{\left(- x + \pi \right)}} + \frac{16563}{1600 \csc{\left(- 2 x + \pi \right)}} + \frac{211}{25 \csc^{2}{\left(- x + \frac{\pi}{2} \right)}}$$
2 2 211 6311*x 2563*sin(x) 422*cos(x) 211*sin (x) 211*cos (x) 16563*cos(x)*sin(x) --- - ------ - ----------- - ---------- - ----------- + ----------- + ------------------- 50 800 200 25 50 50 800
$$- \frac{211 \sin^{2}{\left(x \right)}}{50} + \frac{16563 \sin{\left(x \right)} \cos{\left(x \right)}}{800} + \frac{211 \cos^{2}{\left(x \right)}}{50} - \frac{6311 x}{800} - \frac{2563 \sin{\left(x \right)}}{200} - \frac{422 \cos{\left(x \right)}}{25} + \frac{211}{50}$$
/ 2/x pi\\
2563*|1 - cot |- + --||*(1 + sin(x))
6311*x 422*cos(x) 211*(1 + cos(2*x)) 16563*sin(2*x) \ \2 4 //
- ------ - ---------- + ------------------ + -------------- - ------------------------------------
800 25 50 1600 400
$$- \frac{2563 \cdot \left(- \cot^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1\right) \left(\sin{\left(x \right)} + 1\right)}{400} - \frac{6311 x}{800} + \frac{211 \left(\cos{\left(2 x \right)} + 1\right)}{50} + \frac{16563 \sin{\left(2 x \right)}}{1600} - \frac{422 \cos{\left(x \right)}}{25}$$
/x pi\ 2/x pi\
844*tan|- + --| 844*tan |- + --|
6311*x 2563*sin(x) 16563*sin(2*x) \2 4 / \2 4 /
- ------ - ----------- + -------------- - --------------------- + ----------------------
800 200 1600 / 2/x pi\\ 2
25*|1 + tan |- + --|| / 2/x pi\\
\ \2 4 // 25*|1 + tan |- + --||
\ \2 4 //
$$- \frac{6311 x}{800} - \frac{2563 \sin{\left(x \right)}}{200} + \frac{16563 \sin{\left(2 x \right)}}{1600} - \frac{844 \tan{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}{25 \left(\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1\right)} + \frac{844 \tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}{25 \left(\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}}$$
/x\ / 2/x\\
2563*tan|-| 422*|1 - tan |-|| / 2 \
211 6311*x \2/ \ \2// 211*\1 - tan (x)/ 16563*tan(x)
--- - ------ - ----------------- - ----------------- + ----------------- + -----------------
50 800 / 2/x\\ / 2/x\\ / 2 \ / 2 \
100*|1 + tan |-|| 25*|1 + tan |-|| 50*\1 + tan (x)/ 800*\1 + tan (x)/
\ \2// \ \2//
$$- \frac{6311 x}{800} - \frac{422 \cdot \left(- \tan^{2}{\left(\frac{x}{2} \right)} + 1\right)}{25 \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)} + \frac{211 \cdot \left(- \tan^{2}{\left(x \right)} + 1\right)}{50 \left(\tan^{2}{\left(x \right)} + 1\right)} + \frac{211}{50} + \frac{16563 \tan{\left(x \right)}}{800 \left(\tan^{2}{\left(x \right)} + 1\right)} - \frac{2563 \tan{\left(\frac{x}{2} \right)}}{100 \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)}$$
2
/x\ / 2/x\\ / 2/x\\
2563*tan|-| 422*|1 - tan |-|| 211*|1 - tan |-||
6311*x \2/ \ \2// \ \2// 16563*tan(x)
- ------ - ----------------- - ----------------- + ------------------ + -----------------
800 / 2/x\\ / 2/x\\ 2 / 2 \
100*|1 + tan |-|| 25*|1 + tan |-|| / 2/x\\ 800*\1 + tan (x)/
\ \2// \ \2// 25*|1 + tan |-||
\ \2//
$$- \frac{6311 x}{800} + \frac{211 \left(- \tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}}{25 \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}} - \frac{422 \cdot \left(- \tan^{2}{\left(\frac{x}{2} \right)} + 1\right)}{25 \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)} + \frac{16563 \tan{\left(x \right)}}{800 \left(\tan^{2}{\left(x \right)} + 1\right)} - \frac{2563 \tan{\left(\frac{x}{2} \right)}}{100 \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)}$$
/x\ /x pi\ 2/x pi\
2563*tan|-| 844*tan|- + --| 844*tan |- + --|
6311*x \2/ \2 4 / \2 4 / 16563*tan(x)
- ------ - ----------------- - --------------------- + ---------------------- + -----------------
800 / 2/x\\ / 2/x pi\\ 2 / 2 \
100*|1 + tan |-|| 25*|1 + tan |- + --|| / 2/x pi\\ 800*\1 + tan (x)/
\ \2// \ \2 4 // 25*|1 + tan |- + --||
\ \2 4 //
$$- \frac{6311 x}{800} - \frac{844 \tan{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}{25 \left(\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1\right)} + \frac{844 \tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}{25 \left(\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}} + \frac{16563 \tan{\left(x \right)}}{800 \left(\tan^{2}{\left(x \right)} + 1\right)} - \frac{2563 \tan{\left(\frac{x}{2} \right)}}{100 \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)}$$
/x\ /x pi\ 2/x pi\
2563*cot|-| 844*tan|- + --| 844*tan |- + --|
6311*x \2/ \2 4 / \2 4 / 16563*cot(x)
- ------ - ----------------- - --------------------- + ---------------------- + -----------------
800 / 2/x\\ / 2/x pi\\ 2 / 2 \
100*|1 + cot |-|| 25*|1 + tan |- + --|| / 2/x pi\\ 800*\1 + cot (x)/
\ \2// \ \2 4 // 25*|1 + tan |- + --||
\ \2 4 //
$$- \frac{6311 x}{800} + \frac{16563 \cot{\left(x \right)}}{800 \left(\cot^{2}{\left(x \right)} + 1\right)} - \frac{2563 \cot{\left(\frac{x}{2} \right)}}{100 \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)} - \frac{844 \tan{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}{25 \left(\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1\right)} + \frac{844 \tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}{25 \left(\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}}$$
2
/ 1 \ / 1 \
422*|1 - -------| 211*|1 - -------|
| 2/x\| | 2/x\|
| cot |-|| | cot |-||
6311*x 2563 \ \2// \ \2// 16563
- ------ - ------------------------ - ----------------- + ------------------ + ------------------------
800 / 1 \ /x\ / 1 \ 2 / 1 \
100*|1 + -------|*cot|-| 25*|1 + -------| / 1 \ 800*|1 + -------|*cot(x)
| 2/x\| \2/ | 2/x\| 25*|1 + -------| | 2 |
| cot |-|| | cot |-|| | 2/x\| \ cot (x)/
\ \2// \ \2// | cot |-||
\ \2//
$$- \frac{6311 x}{800} + \frac{211 \left(1 - \frac{1}{\cot^{2}{\left(\frac{x}{2} \right)}}\right)^{2}}{25 \left(1 + \frac{1}{\cot^{2}{\left(\frac{x}{2} \right)}}\right)^{2}} - \frac{422 \cdot \left(1 - \frac{1}{\cot^{2}{\left(\frac{x}{2} \right)}}\right)}{25 \cdot \left(1 + \frac{1}{\cot^{2}{\left(\frac{x}{2} \right)}}\right)} + \frac{16563}{800 \cdot \left(1 + \frac{1}{\cot^{2}{\left(x \right)}}\right) \cot{\left(x \right)}} - \frac{2563}{100 \cdot \left(1 + \frac{1}{\cot^{2}{\left(\frac{x}{2} \right)}}\right) \cot{\left(\frac{x}{2} \right)}}$$
2
/ 2/x pi\\ / 2/x\\ / 2/x\\ / 2/ pi\\
2563*|-1 + tan |- + --|| 422*|-1 + cot |-|| 211*|-1 + cot |-|| 16563*|-1 + tan |x + --||
6311*x \ \2 4 // \ \2// \ \2// \ \ 4 //
- ------ - ------------------------ - ------------------ + ------------------- + -------------------------
800 / 2/x pi\\ / 2/x\\ 2 / 2/ pi\\
200*|1 + tan |- + --|| 25*|1 + cot |-|| / 2/x\\ 1600*|1 + tan |x + --||
\ \2 4 // \ \2// 25*|1 + cot |-|| \ \ 4 //
\ \2//
$$- \frac{6311 x}{800} - \frac{2563 \left(\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} - 1\right)}{200 \left(\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1\right)} + \frac{16563 \left(\tan^{2}{\left(x + \frac{\pi}{4} \right)} - 1\right)}{1600 \left(\tan^{2}{\left(x + \frac{\pi}{4} \right)} + 1\right)} + \frac{211 \left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)^{2}}{25 \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}} - \frac{422 \left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)}{25 \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)}$$
2
/ 2/x pi\\ / 2/x\\ / 2/x\\ / 2/ pi\\
2563*|1 - cot |- + --|| 422*|1 - tan |-|| 211*|1 - tan |-|| 16563*|1 - cot |x + --||
6311*x \ \2 4 // \ \2// \ \2// \ \ 4 //
- ------ - ----------------------- - ----------------- + ------------------ + ------------------------
800 / 2/x pi\\ / 2/x\\ 2 / 2/ pi\\
200*|1 + cot |- + --|| 25*|1 + tan |-|| / 2/x\\ 1600*|1 + cot |x + --||
\ \2 4 // \ \2// 25*|1 + tan |-|| \ \ 4 //
\ \2//
$$- \frac{6311 x}{800} + \frac{211 \left(- \tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}}{25 \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}} - \frac{422 \cdot \left(- \tan^{2}{\left(\frac{x}{2} \right)} + 1\right)}{25 \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)} - \frac{2563 \cdot \left(- \cot^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1\right)}{200 \left(\cot^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1\right)} + \frac{16563 \cdot \left(- \cot^{2}{\left(x + \frac{\pi}{4} \right)} + 1\right)}{1600 \left(\cot^{2}{\left(x + \frac{\pi}{4} \right)} + 1\right)}$$
2
/ 4/x\\
| 4*sin |-||
| \2/|
211*|1 - ---------| 2/x\
| 2 | 2563*sin |-|*sin(x)
6311*x 16563*sin(2*x) 844*(-1 - cos(2*x) + 2*cos(x)) \ sin (x) / \2/
- ------ + -------------- - ----------------------------------- + -------------------- - ------------------------
800 1600 / 2\ 2 / 2 4/x\\
25*\1 - cos(2*x) + 2*(1 - cos(x)) / / 4/x\\ 50*|sin (x) + 4*sin |-||
| 4*sin |-|| \ \2//
| \2/|
25*|1 + ---------|
| 2 |
\ sin (x) /
$$- \frac{2563 \sin^{2}{\left(\frac{x}{2} \right)} \sin{\left(x \right)}}{50 \cdot \left(4 \sin^{4}{\left(\frac{x}{2} \right)} + \sin^{2}{\left(x \right)}\right)} - \frac{6311 x}{800} + \frac{16563 \sin{\left(2 x \right)}}{1600} + \frac{211 \left(- \frac{4 \sin^{4}{\left(\frac{x}{2} \right)}}{\sin^{2}{\left(x \right)}} + 1\right)^{2}}{25 \left(\frac{4 \sin^{4}{\left(\frac{x}{2} \right)}}{\sin^{2}{\left(x \right)}} + 1\right)^{2}} - \frac{844 \cdot \left(2 \cos{\left(x \right)} - \cos{\left(2 x \right)} - 1\right)}{25 \cdot \left(2 \left(- \cos{\left(x \right)} + 1\right)^{2} - \cos{\left(2 x \right)} + 1\right)}$$
// 1 for x mod 2*pi = 0\
|| |
// 0 for x mod pi = 0\ // 1 for x mod 2*pi = 0\ 211*|< 2 | // 0 for 2*x mod pi = 0\
2563*|< | 422*|< | ||cos (x) otherwise | 16563*|< |
6311*x \\sin(x) otherwise / \\cos(x) otherwise / \\ / \\sin(2*x) otherwise /
- ------ - -------------------------------- - --------------------------------- + ---------------------------------- + -------------------------------------
800 200 25 25 1600
$$- \frac{6311 x}{800} - \left(\frac{2563 \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\sin{\left(x \right)} & \text{otherwise} \end{cases}\right)}{200}\right) + \left(\frac{16563 \left(\begin{cases} 0 & \text{for}\: 2 x \bmod \pi = 0 \\\sin{\left(2 x \right)} & \text{otherwise} \end{cases}\right)}{1600}\right) - \left(\frac{422 \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\cos{\left(x \right)} & \text{otherwise} \end{cases}\right)}{25}\right) + \left(\frac{211 \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\cos^{2}{\left(x \right)} & \text{otherwise} \end{cases}\right)}{25}\right)$$
// 0 for x mod pi = 0\ // 1 for x mod 2*pi = 0\ // 1 for x mod pi = 0\ // 0 for 2*x mod pi = 0\
2563*|< | 422*|< | 211*|< | 16563*|< |
211 6311*x \\sin(x) otherwise / \\cos(x) otherwise / \\cos(2*x) otherwise / \\sin(2*x) otherwise /
--- - ------ - -------------------------------- - --------------------------------- + --------------------------------- + -------------------------------------
50 800 200 25 50 1600
$$- \frac{6311 x}{800} - \left(\frac{2563 \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\sin{\left(x \right)} & \text{otherwise} \end{cases}\right)}{200}\right) + \left(\frac{16563 \left(\begin{cases} 0 & \text{for}\: 2 x \bmod \pi = 0 \\\sin{\left(2 x \right)} & \text{otherwise} \end{cases}\right)}{1600}\right) + \left(\frac{211 \left(\begin{cases} 1 & \text{for}\: x \bmod \pi = 0 \\\cos{\left(2 x \right)} & \text{otherwise} \end{cases}\right)}{50}\right) - \left(\frac{422 \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\cos{\left(x \right)} & \text{otherwise} \end{cases}\right)}{25}\right) + \frac{211}{50}$$
// 0 for x mod pi = 0\ // 1 for x mod 2*pi = 0\ // 0 for 2*x mod pi = 0\
|| | || | || |
2563*|< / pi\ | // 1 for x mod 2*pi = 0\ 211*|< 2 | 16563*|< / pi\ |
||cos|x - --| otherwise | 422*|< | ||cos (x) otherwise | ||cos|2*x - --| otherwise |
6311*x \\ \ 2 / / \\cos(x) otherwise / \\ / \\ \ 2 / /
- ------ - ------------------------------------- - --------------------------------- + ---------------------------------- + ------------------------------------------
800 200 25 25 1600
$$- \frac{6311 x}{800} - \left(\frac{2563 \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\cos{\left(x - \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right)}{200}\right) + \left(\frac{16563 \left(\begin{cases} 0 & \text{for}\: 2 x \bmod \pi = 0 \\\cos{\left(2 x - \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right)}{1600}\right) - \left(\frac{422 \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\cos{\left(x \right)} & \text{otherwise} \end{cases}\right)}{25}\right) + \left(\frac{211 \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\cos^{2}{\left(x \right)} & \text{otherwise} \end{cases}\right)}{25}\right)$$
// 1 for x mod 2*pi = 0\ // 1 for x mod 2*pi = 0\
|| | || |
// 0 for x mod pi = 0\ 422*|< / pi\ | 211*|< 2/ pi\ | // 0 for 2*x mod pi = 0\
2563*|< | ||sin|x + --| otherwise | ||sin |x + --| otherwise | 16563*|< |
6311*x \\sin(x) otherwise / \\ \ 2 / / \\ \ 2 / / \\sin(2*x) otherwise /
- ------ - -------------------------------- - -------------------------------------- + --------------------------------------- + -------------------------------------
800 200 25 25 1600
$$- \frac{6311 x}{800} - \left(\frac{2563 \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\sin{\left(x \right)} & \text{otherwise} \end{cases}\right)}{200}\right) + \left(\frac{16563 \left(\begin{cases} 0 & \text{for}\: 2 x \bmod \pi = 0 \\\sin{\left(2 x \right)} & \text{otherwise} \end{cases}\right)}{1600}\right) - \left(\frac{422 \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\sin{\left(x + \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right)}{25}\right) + \left(\frac{211 \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\sin^{2}{\left(x + \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right)}{25}\right)$$
// 0 for x mod pi = 0\
|| |
||1 - cos(x) | // 1 for x mod 2*pi = 0\
2563*|<---------- otherwise | || |
|| /x\ | // 1 for x mod 2*pi = 0\ 211*|< 2 | // 0 for 2*x mod pi = 0\
|| tan|-| | 422*|< | ||cos (x) otherwise | 16563*|< |
6311*x \\ \2/ / \\cos(x) otherwise / \\ / \\sin(2*x) otherwise /
- ------ - ------------------------------------ - --------------------------------- + ---------------------------------- + -------------------------------------
800 200 25 25 1600
$$- \frac{6311 x}{800} - \left(\frac{2563 \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{- \cos{\left(x \right)} + 1}{\tan{\left(\frac{x}{2} \right)}} & \text{otherwise} \end{cases}\right)}{200}\right) + \left(\frac{16563 \left(\begin{cases} 0 & \text{for}\: 2 x \bmod \pi = 0 \\\sin{\left(2 x \right)} & \text{otherwise} \end{cases}\right)}{1600}\right) - \left(\frac{422 \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\cos{\left(x \right)} & \text{otherwise} \end{cases}\right)}{25}\right) + \left(\frac{211 \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\cos^{2}{\left(x \right)} & \text{otherwise} \end{cases}\right)}{25}\right)$$
// / 3*pi\ \ // 1 for x mod 2*pi = 0\ // / 3*pi\ \
|| 1 for |x + ----| mod 2*pi = 0| || | || 1 for |2*x + ----| mod 2*pi = 0|
2563*|< \ 2 / | // 1 for x mod 2*pi = 0\ 211*|< 2 | 16563*|< \ 2 / |
|| | 422*|< | ||cos (x) otherwise | || |
6311*x \\sin(x) otherwise / \\cos(x) otherwise / \\ / \\sin(2*x) otherwise /
- ------ - ------------------------------------------- - --------------------------------- + ---------------------------------- + ------------------------------------------------
800 200 25 25 1600
$$- \frac{6311 x}{800} - \left(\frac{422 \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\cos{\left(x \right)} & \text{otherwise} \end{cases}\right)}{25}\right) + \left(\frac{211 \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\cos^{2}{\left(x \right)} & \text{otherwise} \end{cases}\right)}{25}\right) - \left(\frac{2563 \left(\begin{cases} 1 & \text{for}\: \left(x + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\sin{\left(x \right)} & \text{otherwise} \end{cases}\right)}{200}\right) + \left(\frac{16563 \left(\begin{cases} 1 & \text{for}\: \left(2 x + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\sin{\left(2 x \right)} & \text{otherwise} \end{cases}\right)}{1600}\right)$$
// 0 for x mod pi = 0\ // 1 for x mod 2*pi = 0\ // 0 for 2*x mod pi = 0\
|| | || | || |
|| 1 | // 1 for x mod 2*pi = 0\ || 1 | || 1 |
2563*|<----------- otherwise | || | 211*|<------- otherwise | 16563*|<------------- otherwise |
|| / pi\ | 422*|< 1 | || 2 | || / pi\ |
||sec|x - --| | ||------ otherwise | ||sec (x) | ||sec|2*x - --| |
6311*x \\ \ 2 / / \\sec(x) / \\ / \\ \ 2 / /
- ------ - ------------------------------------- - --------------------------------- + ---------------------------------- + ------------------------------------------
800 200 25 25 1600
$$- \frac{6311 x}{800} - \left(\frac{2563 \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{1}{\sec{\left(x - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)}{200}\right) + \left(\frac{16563 \left(\begin{cases} 0 & \text{for}\: 2 x \bmod \pi = 0 \\\frac{1}{\sec{\left(2 x - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)}{1600}\right) + \left(\frac{211 \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{1}{\sec^{2}{\left(x \right)}} & \text{otherwise} \end{cases}\right)}{25}\right) - \left(\frac{422 \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{1}{\sec{\left(x \right)}} & \text{otherwise} \end{cases}\right)}{25}\right)$$
// 1 for x mod 2*pi = 0\ // 1 for x mod 2*pi = 0\
|| | || |
// 0 for x mod pi = 0\ || 1 | || 1 | // 0 for 2*x mod pi = 0\
|| | 422*|<----------- otherwise | 211*|<------------ otherwise | || |
2563*|< 1 | || /pi \ | || 2/pi \ | 16563*|< 1 |
||------ otherwise | ||csc|-- - x| | ||csc |-- - x| | ||-------- otherwise |
6311*x \\csc(x) / \\ \2 / / \\ \2 / / \\csc(2*x) /
- ------ - -------------------------------- - -------------------------------------- + --------------------------------------- + -------------------------------------
800 200 25 25 1600
$$- \frac{6311 x}{800} - \left(\frac{2563 \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{1}{\csc{\left(x \right)}} & \text{otherwise} \end{cases}\right)}{200}\right) + \left(\frac{16563 \left(\begin{cases} 0 & \text{for}\: 2 x \bmod \pi = 0 \\\frac{1}{\csc{\left(2 x \right)}} & \text{otherwise} \end{cases}\right)}{1600}\right) + \left(\frac{211 \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{1}{\csc^{2}{\left(- x + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)}{25}\right) - \left(\frac{422 \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{1}{\csc{\left(- x + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)}{25}\right)$$
2
/ 4/x\\ / 4/x\\
| 4*sin |-|| | 4*sin |-||
| \2/| | \2/|
422*|1 - ---------| 211*|1 - ---------| 2/x\
| 2 | | 2 | 2563*sin |-| 2
6311*x \ sin (x) / \ sin (x) / \2/ 16563*sin (x)
- ------ - ------------------- + -------------------- - ------------------------- + ----------------------------
800 / 4/x\\ 2 / 4/x\\ / 4 \
| 4*sin |-|| / 4/x\\ | 4*sin |-|| | 4*sin (x)|
| \2/| | 4*sin |-|| | \2/| 400*|1 + ---------|*sin(2*x)
25*|1 + ---------| | \2/| 50*|1 + ---------|*sin(x) | 2 |
| 2 | 25*|1 + ---------| | 2 | \ sin (2*x)/
\ sin (x) / | 2 | \ sin (x) /
\ sin (x) /
$$- \frac{6311 x}{800} + \frac{211 \left(- \frac{4 \sin^{4}{\left(\frac{x}{2} \right)}}{\sin^{2}{\left(x \right)}} + 1\right)^{2}}{25 \left(\frac{4 \sin^{4}{\left(\frac{x}{2} \right)}}{\sin^{2}{\left(x \right)}} + 1\right)^{2}} - \frac{422 \left(- \frac{4 \sin^{4}{\left(\frac{x}{2} \right)}}{\sin^{2}{\left(x \right)}} + 1\right)}{25 \cdot \left(\frac{4 \sin^{4}{\left(\frac{x}{2} \right)}}{\sin^{2}{\left(x \right)}} + 1\right)} + \frac{16563 \sin^{2}{\left(x \right)}}{400 \cdot \left(\frac{4 \sin^{4}{\left(x \right)}}{\sin^{2}{\left(2 x \right)}} + 1\right) \sin{\left(2 x \right)}} - \frac{2563 \sin^{2}{\left(\frac{x}{2} \right)}}{50 \cdot \left(\frac{4 \sin^{4}{\left(\frac{x}{2} \right)}}{\sin^{2}{\left(x \right)}} + 1\right) \sin{\left(x \right)}}$$
// / pi\ \ // / pi\ \
|| 0 for |x + --| mod pi = 0| || 0 for |x + --| mod pi = 0|
|| \ 2 / | || \ 2 / |
422*|< | 211*|< |
// 0 for x mod pi = 0\ || /x pi\ | || 2 2/x pi\ | // 0 for 2*x mod pi = 0\
2563*|< | ||(1 + sin(x))*cot|- + --| otherwise | ||(1 + sin(x)) *cot |- + --| otherwise | 16563*|< |
6311*x \\sin(x) otherwise / \\ \2 4 / / \\ \2 4 / / \\sin(2*x) otherwise /
- ------ - -------------------------------- - -------------------------------------------------------- + ---------------------------------------------------------- + -------------------------------------
800 200 25 25 1600
$$- \frac{6311 x}{800} - \left(\frac{2563 \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\sin{\left(x \right)} & \text{otherwise} \end{cases}\right)}{200}\right) + \left(\frac{16563 \left(\begin{cases} 0 & \text{for}\: 2 x \bmod \pi = 0 \\\sin{\left(2 x \right)} & \text{otherwise} \end{cases}\right)}{1600}\right) - \left(\frac{422 \left(\begin{cases} 0 & \text{for}\: \left(x + \frac{\pi}{2}\right) \bmod \pi = 0 \\\left(\sin{\left(x \right)} + 1\right) \cot{\left(\frac{x}{2} + \frac{\pi}{4} \right)} & \text{otherwise} \end{cases}\right)}{25}\right) + \left(\frac{211 \left(\begin{cases} 0 & \text{for}\: \left(x + \frac{\pi}{2}\right) \bmod \pi = 0 \\\left(\sin{\left(x \right)} + 1\right)^{2} \cot^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} & \text{otherwise} \end{cases}\right)}{25}\right)$$
// 0 for x mod pi = 0\ // 1 for x mod 2*pi = 0\
|| | || |
|| /x\ | || 2/x\ | // 1 for x mod pi = 0\ // 0 for 2*x mod pi = 0\
|| 2*cot|-| | ||-1 + cot |-| | || | || |
2563*|< \2/ | 422*|< \2/ | || 2 | || 2*cot(x) |
||----------- otherwise | ||------------ otherwise | 211*|<-1 + cot (x) | 16563*|<----------- otherwise |
|| 2/x\ | || 2/x\ | ||------------ otherwise | || 2 |
||1 + cot |-| | ||1 + cot |-| | || 2 | ||1 + cot (x) |
211 6311*x \\ \2/ / \\ \2/ / \\1 + cot (x) / \\ /
--- - ------ - ------------------------------------- - --------------------------------------- + ------------------------------------- + ----------------------------------------
50 800 200 25 50 1600
$$- \frac{6311 x}{800} - \left(\frac{2563 \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{x}{2} \right)}}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} & \text{otherwise} \end{cases}\right)}{200}\right) + \left(\frac{16563 \left(\begin{cases} 0 & \text{for}\: 2 x \bmod \pi = 0 \\\frac{2 \cot{\left(x \right)}}{\cot^{2}{\left(x \right)} + 1} & \text{otherwise} \end{cases}\right)}{1600}\right) + \left(\frac{211 \left(\begin{cases} 1 & \text{for}\: x \bmod \pi = 0 \\\frac{\cot^{2}{\left(x \right)} - 1}{\cot^{2}{\left(x \right)} + 1} & \text{otherwise} \end{cases}\right)}{50}\right) - \left(\frac{422 \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{\cot^{2}{\left(\frac{x}{2} \right)} - 1}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} & \text{otherwise} \end{cases}\right)}{25}\right) + \frac{211}{50}$$
// 1 for x mod 2*pi = 0\
|| |
// 0 for x mod pi = 0\ // 1 for x mod 2*pi = 0\ || 2 |
|| | || | ||/ 2/x\\ |
|| /x\ | || 2/x\ | |||-1 + cot |-|| | // 0 for 2*x mod pi = 0\
|| 2*cot|-| | ||-1 + cot |-| | 211*|<\ \2// | || |
2563*|< \2/ | 422*|< \2/ | ||--------------- otherwise | || 2*cot(x) |
||----------- otherwise | ||------------ otherwise | || 2 | 16563*|<----------- otherwise |
|| 2/x\ | || 2/x\ | || / 2/x\\ | || 2 |
||1 + cot |-| | ||1 + cot |-| | || |1 + cot |-|| | ||1 + cot (x) |
6311*x \\ \2/ / \\ \2/ / \\ \ \2// / \\ /
- ------ - ------------------------------------- - --------------------------------------- + ------------------------------------------ + ----------------------------------------
800 200 25 25 1600
$$- \frac{6311 x}{800} - \left(\frac{2563 \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{x}{2} \right)}}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} & \text{otherwise} \end{cases}\right)}{200}\right) + \left(\frac{16563 \left(\begin{cases} 0 & \text{for}\: 2 x \bmod \pi = 0 \\\frac{2 \cot{\left(x \right)}}{\cot^{2}{\left(x \right)} + 1} & \text{otherwise} \end{cases}\right)}{1600}\right) - \left(\frac{422 \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{\cot^{2}{\left(\frac{x}{2} \right)} - 1}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} & \text{otherwise} \end{cases}\right)}{25}\right) + \left(\frac{211 \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{\left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)^{2}}{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right)}{25}\right)$$
// 1 for x mod 2*pi = 0\
|| |
// 0 for x mod pi = 0\ // 1 for x mod 2*pi = 0\ || 2 |
|| | || | ||/ 2/x\\ |
|| /x\ | || 2/x\ | |||1 - tan |-|| | // 0 for 2*x mod pi = 0\
|| 2*tan|-| | ||1 - tan |-| | 211*|<\ \2// | || |
2563*|< \2/ | 422*|< \2/ | ||-------------- otherwise | || 2*tan(x) |
||----------- otherwise | ||----------- otherwise | || 2 | 16563*|<----------- otherwise |
|| 2/x\ | || 2/x\ | ||/ 2/x\\ | || 2 |
||1 + tan |-| | ||1 + tan |-| | |||1 + tan |-|| | ||1 + tan (x) |
6311*x \\ \2/ / \\ \2/ / \\\ \2// / \\ /
- ------ - ------------------------------------- - -------------------------------------- + ----------------------------------------- + ----------------------------------------
800 200 25 25 1600
$$- \frac{6311 x}{800} - \left(\frac{2563 \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{2 \tan{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} & \text{otherwise} \end{cases}\right)}{200}\right) + \left(\frac{16563 \left(\begin{cases} 0 & \text{for}\: 2 x \bmod \pi = 0 \\\frac{2 \tan{\left(x \right)}}{\tan^{2}{\left(x \right)} + 1} & \text{otherwise} \end{cases}\right)}{1600}\right) - \left(\frac{422 \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{- \tan^{2}{\left(\frac{x}{2} \right)} + 1}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} & \text{otherwise} \end{cases}\right)}{25}\right) + \left(\frac{211 \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{\left(- \tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}}{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right)}{25}\right)$$
2
/ 2/x\ \ / 2/x\ \
| sec |-| | | sec |-| |
| \2/ | | \2/ |
422*|1 - ------------| 211*|1 - ------------|
| 2/x pi\| | 2/x pi\| /x\
| sec |- - --|| | sec |- - --|| 2563*sec|-|
6311*x \ \2 2 // \ \2 2 // \2/ 16563*sec(x)
- ------ - ---------------------- + ----------------------- - ---------------------------------- + ----------------------------------
800 / 2/x\ \ 2 / 2/x\ \ / 2 \
| sec |-| | / 2/x\ \ | sec |-| | | sec (x) | / pi\
| \2/ | | sec |-| | | \2/ | /x pi\ 800*|1 + ------------|*sec|x - --|
25*|1 + ------------| | \2/ | 100*|1 + ------------|*sec|- - --| | 2/ pi\| \ 2 /
| 2/x pi\| 25*|1 + ------------| | 2/x pi\| \2 2 / | sec |x - --||
| sec |- - --|| | 2/x pi\| | sec |- - --|| \ \ 2 //
\ \2 2 // | sec |- - --|| \ \2 2 //
\ \2 2 //
$$- \frac{6311 x}{800} + \frac{211 \left(- \frac{\sec^{2}{\left(\frac{x}{2} \right)}}{\sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} + 1\right)^{2}}{25 \left(\frac{\sec^{2}{\left(\frac{x}{2} \right)}}{\sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} + 1\right)^{2}} - \frac{422 \left(- \frac{\sec^{2}{\left(\frac{x}{2} \right)}}{\sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} + 1\right)}{25 \left(\frac{\sec^{2}{\left(\frac{x}{2} \right)}}{\sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} + 1\right)} + \frac{16563 \sec{\left(x \right)}}{800 \left(\frac{\sec^{2}{\left(x \right)}}{\sec^{2}{\left(x - \frac{\pi}{2} \right)}} + 1\right) \sec{\left(x - \frac{\pi}{2} \right)}} - \frac{2563 \sec{\left(\frac{x}{2} \right)}}{100 \left(\frac{\sec^{2}{\left(\frac{x}{2} \right)}}{\sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} + 1\right) \sec{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}$$
2
/ 2/x pi\\ / 2/x pi\\
| cos |- - --|| | cos |- - --||
| \2 2 /| | \2 2 /|
422*|1 - ------------| 211*|1 - ------------|
| 2/x\ | | 2/x\ | /x pi\ / pi\
| cos |-| | | cos |-| | 2563*cos|- - --| 16563*cos|x - --|
6311*x \ \2/ / \ \2/ / \2 2 / \ 2 /
- ------ - ---------------------- + ----------------------- - ----------------------------- + -----------------------------
800 / 2/x pi\\ 2 / 2/x pi\\ / 2/ pi\\
| cos |- - --|| / 2/x pi\\ | cos |- - --|| | cos |x - --||
| \2 2 /| | cos |- - --|| | \2 2 /| /x\ | \ 2 /|
25*|1 + ------------| | \2 2 /| 100*|1 + ------------|*cos|-| 800*|1 + ------------|*cos(x)
| 2/x\ | 25*|1 + ------------| | 2/x\ | \2/ | 2 |
| cos |-| | | 2/x\ | | cos |-| | \ cos (x) /
\ \2/ / | cos |-| | \ \2/ /
\ \2/ /
$$- \frac{6311 x}{800} + \frac{211 \left(1 - \frac{\cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{x}{2} \right)}}\right)^{2}}{25 \left(1 + \frac{\cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{x}{2} \right)}}\right)^{2}} - \frac{422 \cdot \left(1 - \frac{\cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{x}{2} \right)}}\right)}{25 \cdot \left(1 + \frac{\cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{x}{2} \right)}}\right)} + \frac{16563 \cos{\left(x - \frac{\pi}{2} \right)}}{800 \cdot \left(1 + \frac{\cos^{2}{\left(x - \frac{\pi}{2} \right)}}{\cos^{2}{\left(x \right)}}\right) \cos{\left(x \right)}} - \frac{2563 \cos{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{100 \cdot \left(1 + \frac{\cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{x}{2} \right)}}\right) \cos{\left(\frac{x}{2} \right)}}$$
2
/ 2/pi x\\ / 2/pi x\\
| csc |-- - -|| | csc |-- - -||
| \2 2/| | \2 2/|
422*|1 - ------------| 211*|1 - ------------|
| 2/x\ | | 2/x\ | /pi x\ /pi \
| csc |-| | | csc |-| | 2563*csc|-- - -| 16563*csc|-- - x|
6311*x \ \2/ / \ \2/ / \2 2/ \2 /
- ------ - ---------------------- + ----------------------- - ----------------------------- + -----------------------------
800 / 2/pi x\\ 2 / 2/pi x\\ / 2/pi \\
| csc |-- - -|| / 2/pi x\\ | csc |-- - -|| | csc |-- - x||
| \2 2/| | csc |-- - -|| | \2 2/| /x\ | \2 /|
25*|1 + ------------| | \2 2/| 100*|1 + ------------|*csc|-| 800*|1 + ------------|*csc(x)
| 2/x\ | 25*|1 + ------------| | 2/x\ | \2/ | 2 |
| csc |-| | | 2/x\ | | csc |-| | \ csc (x) /
\ \2/ / | csc |-| | \ \2/ /
\ \2/ /
$$- \frac{6311 x}{800} + \frac{211 \left(1 - \frac{\csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{x}{2} \right)}}\right)^{2}}{25 \left(1 + \frac{\csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{x}{2} \right)}}\right)^{2}} - \frac{422 \cdot \left(1 - \frac{\csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{x}{2} \right)}}\right)}{25 \cdot \left(1 + \frac{\csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{x}{2} \right)}}\right)} + \frac{16563 \csc{\left(- x + \frac{\pi}{2} \right)}}{800 \cdot \left(1 + \frac{\csc^{2}{\left(- x + \frac{\pi}{2} \right)}}{\csc^{2}{\left(x \right)}}\right) \csc{\left(x \right)}} - \frac{2563 \csc{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}{100 \cdot \left(1 + \frac{\csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{x}{2} \right)}}\right) \csc{\left(\frac{x}{2} \right)}}$$
// 1 for x mod 2*pi = 0\
|| |
// 0 for x mod pi = 0\ // 1 for x mod 2*pi = 0\ ||/ 1 for x mod 2*pi = 0 | // 0 for 2*x mod pi = 0\
|| | || | 211*|<| | || |
2563*|
$$- \frac{6311 x}{800} - \left(\frac{2563 \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\sin{\left(x \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)}{200}\right) + \left(\frac{16563 \left(\begin{cases} 0 & \text{for}\: 2 x \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: 2 x \bmod \pi = 0 \\\sin{\left(2 x \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)}{1600}\right) - \left(\frac{422 \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\cos{\left(x \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)}{25}\right) + \left(\frac{211 \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\cos^{2}{\left(x \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)}{25}\right)$$
// 1 for x mod 2*pi = 0\
|| |
// 1 for x mod 2*pi = 0\ || 2 |
|| | ||/ 1 \ |
|| 1 | |||-1 + -------| |
||-1 + ------- | ||| 2/x\| |
// 0 for x mod pi = 0\ || 2/x\ | ||| tan |-|| | // 0 for 2*x mod pi = 0\
|| | || tan |-| | 211*|<\ \2// | || |
|| 2 | 422*|< \2/ | ||--------------- otherwise | || 2 |
||-------------------- otherwise | ||------------ otherwise | || 2 | ||-------------------- otherwise |
2563*|
$$- \frac{6311 x}{800} - \left(\frac{2563 \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{2}{\left(1 + \frac{1}{\tan^{2}{\left(\frac{x}{2} \right)}}\right) \tan{\left(\frac{x}{2} \right)}} & \text{otherwise} \end{cases}\right)}{200}\right) + \left(\frac{16563 \left(\begin{cases} 0 & \text{for}\: 2 x \bmod \pi = 0 \\\frac{2}{\left(1 + \frac{1}{\tan^{2}{\left(x \right)}}\right) \tan{\left(x \right)}} & \text{otherwise} \end{cases}\right)}{1600}\right) - \left(\frac{422 \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{-1 + \frac{1}{\tan^{2}{\left(\frac{x}{2} \right)}}}{1 + \frac{1}{\tan^{2}{\left(\frac{x}{2} \right)}}} & \text{otherwise} \end{cases}\right)}{25}\right) + \left(\frac{211 \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{\left(-1 + \frac{1}{\tan^{2}{\left(\frac{x}{2} \right)}}\right)^{2}}{\left(1 + \frac{1}{\tan^{2}{\left(\frac{x}{2} \right)}}\right)^{2}} & \text{otherwise} \end{cases}\right)}{25}\right)$$
// / pi\ \
|| 0 for |x + --| mod pi = 0|
// / pi\ \ || \ 2 / |
|| 0 for |x + --| mod pi = 0| || |
// 0 for x mod pi = 0\ || \ 2 / | || 2/x pi\ |
|| | || | || 4*cot |- + --| |
|| /x\ | || /x pi\ | 211*|< \2 4 / | // 0 for 2*x mod pi = 0\
|| 2*cot|-| | 422*|< 2*cot|- + --| | ||------------------- otherwise | || |
2563*|< \2/ | || \2 4 / | || 2 | || 2*cot(x) |
||----------- otherwise | ||---------------- otherwise | ||/ 2/x pi\\ | 16563*|<----------- otherwise |
|| 2/x\ | || 2/x pi\ | |||1 + cot |- + --|| | || 2 |
||1 + cot |-| | ||1 + cot |- + --| | ||\ \2 4 // | ||1 + cot (x) |
6311*x \\ \2/ / \\ \2 4 / / \\ / \\ /
- ------ - ------------------------------------- - ------------------------------------------------ + --------------------------------------------------- + ----------------------------------------
800 200 25 25 1600
$$- \frac{6311 x}{800} - \left(\frac{2563 \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{x}{2} \right)}}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} & \text{otherwise} \end{cases}\right)}{200}\right) + \left(\frac{16563 \left(\begin{cases} 0 & \text{for}\: 2 x \bmod \pi = 0 \\\frac{2 \cot{\left(x \right)}}{\cot^{2}{\left(x \right)} + 1} & \text{otherwise} \end{cases}\right)}{1600}\right) + \left(\frac{211 \left(\begin{cases} 0 & \text{for}\: \left(x + \frac{\pi}{2}\right) \bmod \pi = 0 \\\frac{4 \cot^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}{\left(\cot^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right)}{25}\right) - \left(\frac{422 \left(\begin{cases} 0 & \text{for}\: \left(x + \frac{\pi}{2}\right) \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}{\cot^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1} & \text{otherwise} \end{cases}\right)}{25}\right)$$
// / 3*pi\ \ // 1 for x mod 2*pi = 0\ // / 3*pi\ \
|| 1 for |x + ----| mod 2*pi = 0| || | || 1 for |2*x + ----| mod 2*pi = 0|
|| \ 2 / | // 1 for x mod 2*pi = 0\ || 2 | || \ 2 / |
|| | || | ||/ 2/x\\ | || |
|| 2/x pi\ | || 2/x\ | |||-1 + cot |-|| | || 2/ pi\ |
2563*|<-1 + tan |- + --| | ||-1 + cot |-| | 211*|<\ \2// | 16563*|<-1 + tan |x + --| |
|| \2 4 / | 422*|< \2/ | ||--------------- otherwise | || \ 4 / |
||----------------- otherwise | ||------------ otherwise | || 2 | ||----------------- otherwise |
|| 2/x pi\ | || 2/x\ | || / 2/x\\ | || 2/ pi\ |
|| 1 + tan |- + --| | ||1 + cot |-| | || |1 + cot |-|| | || 1 + tan |x + --| |
6311*x \\ \2 4 / / \\ \2/ / \\ \ \2// / \\ \ 4 / /
- ------ - ------------------------------------------------------ - --------------------------------------- + ------------------------------------------ + ---------------------------------------------------------
800 200 25 25 1600
$$- \frac{6311 x}{800} - \left(\frac{422 \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{\cot^{2}{\left(\frac{x}{2} \right)} - 1}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} & \text{otherwise} \end{cases}\right)}{25}\right) + \left(\frac{211 \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{\left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)^{2}}{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right)}{25}\right) - \left(\frac{2563 \left(\begin{cases} 1 & \text{for}\: \left(x + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\frac{\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} - 1}{\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1} & \text{otherwise} \end{cases}\right)}{200}\right) + \left(\frac{16563 \left(\begin{cases} 1 & \text{for}\: \left(2 x + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\frac{\tan^{2}{\left(x + \frac{\pi}{4} \right)} - 1}{\tan^{2}{\left(x + \frac{\pi}{4} \right)} + 1} & \text{otherwise} \end{cases}\right)}{1600}\right)$$
// 0 for x mod pi = 0\ // 1 for x mod 2*pi = 0\
|| | || |
|| 2*sin(x) | || 2 |
||---------------------------- otherwise | ||/ 2 4/x\\ |
|| / 2 \ | // 1 for x mod 2*pi = 0\ |||sin (x) - 4*sin |-|| |
2563*|< | sin (x) | | || | 211*|<\ \2// |
||(1 - cos(x))*|1 + ---------| | || 2 | ||---------------------- otherwise |
|| | 4/x\| | 422*|< -4 + 4*sin (x) + 4*cos(x) | || 2 |
|| | 4*sin |-|| | ||--------------------------- otherwise | ||/ 2 4/x\\ | // 0 for 2*x mod pi = 0\
|| \ \2// | || 2 2 | |||sin (x) + 4*sin |-|| | 16563*|< |
6311*x \\ / \\2*(1 - cos(x)) + 2*sin (x) / \\\ \2// / \\sin(2*x) otherwise /
- ------ - ------------------------------------------------------ - ------------------------------------------------------ + ------------------------------------------------- + -------------------------------------
800 200 25 25 1600
$$- \frac{6311 x}{800} - \left(\frac{2563 \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{2 \sin{\left(x \right)}}{\left(1 + \frac{\sin^{2}{\left(x \right)}}{4 \sin^{4}{\left(\frac{x}{2} \right)}}\right) \left(- \cos{\left(x \right)} + 1\right)} & \text{otherwise} \end{cases}\right)}{200}\right) + \left(\frac{16563 \left(\begin{cases} 0 & \text{for}\: 2 x \bmod \pi = 0 \\\sin{\left(2 x \right)} & \text{otherwise} \end{cases}\right)}{1600}\right) - \left(\frac{422 \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{4 \sin^{2}{\left(x \right)} + 4 \cos{\left(x \right)} - 4}{2 \left(- \cos{\left(x \right)} + 1\right)^{2} + 2 \sin^{2}{\left(x \right)}} & \text{otherwise} \end{cases}\right)}{25}\right) + \left(\frac{211 \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{\left(- 4 \sin^{4}{\left(\frac{x}{2} \right)} + \sin^{2}{\left(x \right)}\right)^{2}}{\left(4 \sin^{4}{\left(\frac{x}{2} \right)} + \sin^{2}{\left(x \right)}\right)^{2}} & \text{otherwise} \end{cases}\right)}{25}\right)$$
// 1 for x mod 2*pi = 0\
|| |
// 1 for x mod 2*pi = 0\ || 2 |
|| | ||/ 2 \ |
|| 2 | ||| sin (x) | |
|| sin (x) | |||-1 + ---------| |
// 0 for x mod pi = 0\ ||-1 + --------- | ||| 4/x\| |
|| | || 4/x\ | ||| 4*sin |-|| |
|| sin(x) | || 4*sin |-| | 211*|<\ \2// | // 0 for 2*x mod pi = 0\
||----------------------- otherwise | 422*|< \2/ | ||----------------- otherwise | || |
||/ 2 \ | ||-------------- otherwise | || 2 | || sin(2*x) |
2563*|<| sin (x) | 2/x\ | || 2 | || / 2 \ | ||----------------------- otherwise |
|||1 + ---------|*sin |-| | || sin (x) | || | sin (x) | | 16563*|
$$- \frac{6311 x}{800} - \left(\frac{2563 \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{\sin{\left(x \right)}}{\left(1 + \frac{\sin^{2}{\left(x \right)}}{4 \sin^{4}{\left(\frac{x}{2} \right)}}\right) \sin^{2}{\left(\frac{x}{2} \right)}} & \text{otherwise} \end{cases}\right)}{200}\right) + \left(\frac{16563 \left(\begin{cases} 0 & \text{for}\: 2 x \bmod \pi = 0 \\\frac{\sin{\left(2 x \right)}}{\left(1 + \frac{\sin^{2}{\left(2 x \right)}}{4 \sin^{4}{\left(x \right)}}\right) \sin^{2}{\left(x \right)}} & \text{otherwise} \end{cases}\right)}{1600}\right) - \left(\frac{422 \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{-1 + \frac{\sin^{2}{\left(x \right)}}{4 \sin^{4}{\left(\frac{x}{2} \right)}}}{1 + \frac{\sin^{2}{\left(x \right)}}{4 \sin^{4}{\left(\frac{x}{2} \right)}}} & \text{otherwise} \end{cases}\right)}{25}\right) + \left(\frac{211 \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{\left(-1 + \frac{\sin^{2}{\left(x \right)}}{4 \sin^{4}{\left(\frac{x}{2} \right)}}\right)^{2}}{\left(1 + \frac{\sin^{2}{\left(x \right)}}{4 \sin^{4}{\left(\frac{x}{2} \right)}}\right)^{2}} & \text{otherwise} \end{cases}\right)}{25}\right)$$
// 1 for x mod 2*pi = 0\
|| |
// 0 for x mod pi = 0\ // 1 for x mod 2*pi = 0\ ||/ 1 for x mod 2*pi = 0 |
|| | || | ||| |
||/ 0 for x mod pi = 0 | ||/ 1 for x mod 2*pi = 0 | ||| 2 | // 0 for 2*x mod pi = 0\
||| | ||| | |||/ 2/x\\ | || |
||| /x\ | ||| 2/x\ | 211*|<||-1 + cot |-|| | ||/ 0 for 2*x mod pi = 0 |
2563*|<| 2*cot|-| | 422*|<|-1 + cot |-| | ||<\ \2// otherwise | ||| |
||< \2/ otherwise | ||< \2/ otherwise | |||--------------- otherwise | 16563*|<| 2*cot(x) |
|||----------- otherwise | |||------------ otherwise | ||| 2 | ||<----------- otherwise otherwise |
||| 2/x\ | ||| 2/x\ | ||| / 2/x\\ | ||| 2 |
|||1 + cot |-| | |||1 + cot |-| | ||| |1 + cot |-|| | |||1 + cot (x) |
6311*x \\\ \2/ / \\\ \2/ / \\\ \ \2// / \\\ /
- ------ - -------------------------------------------------------- - ------------------------------------------------------------ + --------------------------------------------------------------- + -------------------------------------------------------------
800 200 25 25 1600
$$- \frac{6311 x}{800} - \left(\frac{2563 \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{x}{2} \right)}}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)}{200}\right) + \left(\frac{16563 \left(\begin{cases} 0 & \text{for}\: 2 x \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: 2 x \bmod \pi = 0 \\\frac{2 \cot{\left(x \right)}}{\cot^{2}{\left(x \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)}{1600}\right) - \left(\frac{422 \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{\cot^{2}{\left(\frac{x}{2} \right)} - 1}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)}{25}\right) + \left(\frac{211 \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{\left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)^{2}}{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)}{25}\right)$$
// 1 for x mod 2*pi = 0\
|| |
// 1 for x mod 2*pi = 0\ || 2 |
|| | ||/ 2/x\ \ |
|| 2/x\ | ||| cos |-| | |
|| cos |-| | ||| \2/ | |
// 0 for x mod pi = 0\ || \2/ | |||-1 + ------------| |
|| | ||-1 + ------------ | ||| 2/x pi\| |
|| /x\ | || 2/x pi\ | ||| cos |- - --|| | // 0 for 2*x mod pi = 0\
|| 2*cos|-| | || cos |- - --| | 211*|<\ \2 2 // | || |
|| \2/ | 422*|< \2 2 / | ||-------------------- otherwise | || 2*cos(x) |
||------------------------------ otherwise | ||----------------- otherwise | || 2 | ||------------------------------ otherwise |
2563*|
$$- \frac{6311 x}{800} - \left(\frac{2563 \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{2 \cos{\left(\frac{x}{2} \right)}}{\left(\frac{\cos^{2}{\left(\frac{x}{2} \right)}}{\cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} + 1\right) \cos{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)}{200}\right) + \left(\frac{16563 \left(\begin{cases} 0 & \text{for}\: 2 x \bmod \pi = 0 \\\frac{2 \cos{\left(x \right)}}{\left(\frac{\cos^{2}{\left(x \right)}}{\cos^{2}{\left(x - \frac{\pi}{2} \right)}} + 1\right) \cos{\left(x - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)}{1600}\right) - \left(\frac{422 \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{\frac{\cos^{2}{\left(\frac{x}{2} \right)}}{\cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} - 1}{\frac{\cos^{2}{\left(\frac{x}{2} \right)}}{\cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} + 1} & \text{otherwise} \end{cases}\right)}{25}\right) + \left(\frac{211 \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{\left(\frac{\cos^{2}{\left(\frac{x}{2} \right)}}{\cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} - 1\right)^{2}}{\left(\frac{\cos^{2}{\left(\frac{x}{2} \right)}}{\cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} + 1\right)^{2}} & \text{otherwise} \end{cases}\right)}{25}\right)$$
// 1 for x mod 2*pi = 0\
|| |
// 1 for x mod 2*pi = 0\ || 2 |
|| | ||/ 2/x pi\\ |
|| 2/x pi\ | ||| sec |- - --|| |
|| sec |- - --| | ||| \2 2 /| |
// 0 for x mod pi = 0\ || \2 2 / | |||-1 + ------------| | // 0 for 2*x mod pi = 0\
|| | ||-1 + ------------ | ||| 2/x\ | | || |
|| /x pi\ | || 2/x\ | ||| sec |-| | | || / pi\ |
|| 2*sec|- - --| | || sec |-| | 211*|<\ \2/ / | || 2*sec|x - --| |
|| \2 2 / | 422*|< \2/ | ||-------------------- otherwise | || \ 2 / |
||------------------------- otherwise | ||----------------- otherwise | || 2 | ||------------------------- otherwise |
2563*|
$$- \frac{6311 x}{800} - \left(\frac{2563 \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{2 \sec{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\left(1 + \frac{\sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{x}{2} \right)}}\right) \sec{\left(\frac{x}{2} \right)}} & \text{otherwise} \end{cases}\right)}{200}\right) + \left(\frac{16563 \left(\begin{cases} 0 & \text{for}\: 2 x \bmod \pi = 0 \\\frac{2 \sec{\left(x - \frac{\pi}{2} \right)}}{\left(1 + \frac{\sec^{2}{\left(x - \frac{\pi}{2} \right)}}{\sec^{2}{\left(x \right)}}\right) \sec{\left(x \right)}} & \text{otherwise} \end{cases}\right)}{1600}\right) - \left(\frac{422 \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{-1 + \frac{\sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{x}{2} \right)}}}{1 + \frac{\sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{x}{2} \right)}}} & \text{otherwise} \end{cases}\right)}{25}\right) + \left(\frac{211 \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{\left(-1 + \frac{\sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{x}{2} \right)}}\right)^{2}}{\left(1 + \frac{\sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{x}{2} \right)}}\right)^{2}} & \text{otherwise} \end{cases}\right)}{25}\right)$$
// 1 for x mod 2*pi = 0\
|| |
// 1 for x mod 2*pi = 0\ || 2 |
|| | ||/ 2/x\ \ |
|| 2/x\ | ||| csc |-| | |
|| csc |-| | ||| \2/ | |
// 0 for x mod pi = 0\ || \2/ | |||-1 + ------------| |
|| | ||-1 + ------------ | ||| 2/pi x\| |
|| /x\ | || 2/pi x\ | ||| csc |-- - -|| | // 0 for 2*x mod pi = 0\
|| 2*csc|-| | || csc |-- - -| | 211*|<\ \2 2// | || |
|| \2/ | 422*|< \2 2/ | ||-------------------- otherwise | || 2*csc(x) |
||------------------------------ otherwise | ||----------------- otherwise | || 2 | ||------------------------------ otherwise |
2563*|
$$- \frac{6311 x}{800} - \left(\frac{2563 \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{2 \csc{\left(\frac{x}{2} \right)}}{\left(\frac{\csc^{2}{\left(\frac{x}{2} \right)}}{\csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}} + 1\right) \csc{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)}{200}\right) + \left(\frac{16563 \left(\begin{cases} 0 & \text{for}\: 2 x \bmod \pi = 0 \\\frac{2 \csc{\left(x \right)}}{\left(\frac{\csc^{2}{\left(x \right)}}{\csc^{2}{\left(- x + \frac{\pi}{2} \right)}} + 1\right) \csc{\left(- x + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)}{1600}\right) - \left(\frac{422 \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{\frac{\csc^{2}{\left(\frac{x}{2} \right)}}{\csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}} - 1}{\frac{\csc^{2}{\left(\frac{x}{2} \right)}}{\csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}} + 1} & \text{otherwise} \end{cases}\right)}{25}\right) + \left(\frac{211 \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{\left(\frac{\csc^{2}{\left(\frac{x}{2} \right)}}{\csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}} - 1\right)^{2}}{\left(\frac{\csc^{2}{\left(\frac{x}{2} \right)}}{\csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}} + 1\right)^{2}} & \text{otherwise} \end{cases}\right)}{25}\right)$$
-6311*x/800 - 2563*Piecewise((0, Mod(x = pi, 0)), (2*csc(x/2)/((1 + csc(x/2)^2/csc(pi/2 - x/2)^2)*csc(pi/2 - x/2)), True))/200 - 422*Piecewise((1, Mod(x = 2*pi, 0)), ((-1 + csc(x/2)^2/csc(pi/2 - x/2)^2)/(1 + csc(x/2)^2/csc(pi/2 - x/2)^2), True))/25 + 211*Piecewise((1, Mod(x = 2*pi, 0)), ((-1 + csc(x/2)^2/csc(pi/2 - x/2)^2)^2/(1 + csc(x/2)^2/csc(pi/2 - x/2)^2)^2, True))/25 + 16563*Piecewise((0, Mod(2*x = pi, 0)), (2*csc(x)/((1 + csc(x)^2/csc(pi/2 - x)^2)*csc(pi/2 - x)), True))/1600
Рациональный знаменатель
[src]
2 6311*x 2563*sin(x) 422*cos(x) 211*cos (x) 16563*sin(2*x) - ------ - ----------- - ---------- + ----------- + -------------- 800 200 25 25 1600
$$\frac{211 \cos^{2}{\left(x \right)}}{25} - \frac{6311 x}{800} - \frac{2563 \sin{\left(x \right)}}{200} + \frac{16563 \sin{\left(2 x \right)}}{1600} - \frac{422 \cos{\left(x \right)}}{25}$$
2
-216064000000*cos(x) - 164032000000*sin(x) - 100976000000*x + 108032000000*cos (x) + 132504000000*sin(2*x)
----------------------------------------------------------------------------------------------------------
12800000000
$$\frac{108032000000 \cos^{2}{\left(x \right)} - 100976000000 x - 164032000000 \sin{\left(x \right)} + 132504000000 \sin{\left(2 x \right)} - 216064000000 \cos{\left(x \right)}}{12800000000}$$
(-216064000000*cos(x) - 164032000000*sin(x) - 100976000000*x + 108032000000*cos(x)^2 + 132504000000*sin(2*x))/12800000000
Численный ответ
[src]
10.351875*sin(2*x) + 8.44*cos(x)^2 - 7.88875*x - 16.88*cos(x) - 12.815*sin(x)
10.351875*sin(2*x) + 8.44*cos(x)^2 - 7.88875*x - 16.88*cos(x) - 12.815*sin(x)
Степени
[src]
2 6311*x 2563*sin(x) 422*cos(x) 211*cos (x) 16563*sin(2*x) - ------ - ----------- - ---------- + ----------- + -------------- 800 200 25 25 1600
$$\frac{211 \cos^{2}{\left(x \right)}}{25} - \frac{6311 x}{800} - \frac{2563 \sin{\left(x \right)}}{200} + \frac{16563 \sin{\left(2 x \right)}}{1600} - \frac{422 \cos{\left(x \right)}}{25}$$
2
/ I*x -I*x\
|e e |
I*x -I*x 211*|---- + -----| / -2*I*x 2*I*x\ / -I*x I*x\
6311*x 211*e 211*e \ 2 2 / 16563*I*\- e + e / 2563*I*\- e + e /
- ------ - -------- - --------- + ------------------- - ---------------------------- + -----------------------
800 25 25 25 3200 400
$$\frac{211 \left(\frac{e^{i x}}{2} + \frac{e^{- i x}}{2}\right)^{2}}{25} - \frac{6311 x}{800} + \frac{2563 i \left(e^{i x} - e^{- i x}\right)}{400} - \frac{16563 i \left(e^{2 i x} - e^{- 2 i x}\right)}{3200} - \frac{211 e^{i x}}{25} - \frac{211 e^{- i x}}{25}$$
-6311*x/800 - 211*exp(i*x)/25 - 211*exp(-i*x)/25 + 211*(exp(i*x)/2 + exp(-i*x)/2)^2/25 - 16563*i*(-exp(-2*i*x) + exp(2*i*x))/3200 + 2563*i*(-exp(-i*x) + exp(i*x))/400