Господин Экзамен
14*cos(21*x)^2+11*sin(21*x)^2 если x=-1/4
Решение
Вы ввели
[src]
2 2 14*cos (21*x) + 11*sin (21*x)
$$11 \sin^{2}{\left(21 x \right)} + 14 \cos^{2}{\left(21 x \right)}$$
14*cos(21*x)^2 + 11*sin(21*x)^2
Общее упрощение
[src]
2 14 - 3*sin (21*x)
$$- 3 \sin^{2}{\left(21 x \right)} + 14$$
14 - 3*sin(21*x)^2
Подстановка условия
[src]
14*cos(21*x)^2 + 11*sin(21*x)^2 при x = -1/4
подставляем
2 2 14*cos (21*x) + 11*sin (21*x)
$$11 \sin^{2}{\left(21 x \right)} + 14 \cos^{2}{\left(21 x \right)}$$
2 14 - 3*sin (21*x)
$$- 3 \sin^{2}{\left(21 x \right)} + 14$$
переменные
x = -1/4
$$x = - \frac{1}{4}$$
2 14 - 3*sin (21*(-1/4))
$$- 3 \sin^{2}{\left(21 (-1/4) \right)} + 14$$
2 14 - 3*sin (21/4)
$$- 3 \sin^{2}{\left(\frac{21}{4} \right)} + 14$$
14 - 3*sin(21/4)^2
Собрать выражение
[src]
25 3*cos(42*x) -- + ----------- 2 2
$$\frac{3 \cos{\left(42 x \right)}}{2} + \frac{25}{2}$$
25/2 + 3*cos(42*x)/2
Тригонометрическая часть
[src]
2 14 - 3*sin (21*x)
$$- 3 \sin^{2}{\left(21 x \right)} + 14$$
3
14 - ----------
2
csc (21*x)
$$14 - \frac{3}{\csc^{2}{\left(21 x \right)}}$$
25 3*cos(42*x) -- + ----------- 2 2
$$\frac{3 \cos{\left(42 x \right)}}{2} + \frac{25}{2}$$
2/ pi\
14 - 3*cos |21*x - --|
\ 2 /
$$- 3 \cos^{2}{\left(21 x - \frac{\pi}{2} \right)} + 14$$
3
14 - ---------------
2/ pi\
sec |21*x - --|
\ 2 /
$$14 - \frac{3}{\sec^{2}{\left(21 x - \frac{\pi}{2} \right)}}$$
2 2/pi \
11*sin (21*x) + 14*sin |-- + 21*x|
\2 /
$$11 \sin^{2}{\left(21 x \right)} + 14 \sin^{2}{\left(21 x + \frac{\pi}{2} \right)}$$
2/ pi\ 2
11*cos |21*x - --| + 14*cos (21*x)
\ 2 /
$$14 \cos^{2}{\left(21 x \right)} + 11 \cos^{2}{\left(21 x - \frac{\pi}{2} \right)}$$
11 14 ---------- + ---------- 2 2 csc (21*x) sec (21*x)
$$\frac{14}{\sec^{2}{\left(21 x \right)}} + \frac{11}{\csc^{2}{\left(21 x \right)}}$$
11 14
--------------- + ----------
2/ pi\ 2
sec |21*x - --| sec (21*x)
\ 2 /
$$\frac{11}{\sec^{2}{\left(21 x - \frac{\pi}{2} \right)}} + \frac{14}{\sec^{2}{\left(21 x \right)}}$$
11 14
--------------- + ----------
2/pi \ 2
sec |-- - 21*x| sec (21*x)
\2 /
$$\frac{11}{\sec^{2}{\left(- 21 x + \frac{\pi}{2} \right)}} + \frac{14}{\sec^{2}{\left(21 x \right)}}$$
11 14
---------- + ---------------
2 2/pi \
csc (21*x) csc |-- - 21*x|
\2 /
$$\frac{14}{\csc^{2}{\left(- 21 x + \frac{\pi}{2} \right)}} + \frac{11}{\csc^{2}{\left(21 x \right)}}$$
2 2 25 3*sin (21*x) 3*cos (21*x) -- - ------------ + ------------ 2 2 2
$$- \frac{3 \sin^{2}{\left(21 x \right)}}{2} + \frac{3 \cos^{2}{\left(21 x \right)}}{2} + \frac{25}{2}$$
11 14
--------------- + ---------------
2 2/pi \
csc (pi - 21*x) csc |-- - 21*x|
\2 /
$$\frac{14}{\csc^{2}{\left(- 21 x + \frac{\pi}{2} \right)}} + \frac{11}{\csc^{2}{\left(- 21 x + \pi \right)}}$$
2/21*x\
12*tan |----|
\ 2 /
14 - -----------------
2
/ 2/21*x\\
|1 + tan |----||
\ \ 2 //
$$14 - \frac{12 \tan^{2}{\left(\frac{21 x}{2} \right)}}{\left(\tan^{2}{\left(\frac{21 x}{2} \right)} + 1\right)^{2}}$$
2 29 - 22*cos(21*x) - 11*(1 - cos(21*x)) + 7*cos(42*x)
$$- 11 \left(- \cos{\left(21 x \right)} + 1\right)^{2} - 22 \cos{\left(21 x \right)} + 7 \cos{\left(42 x \right)} + 29$$
// 0 for 21*x mod pi = 0\
|| |
14 - 3*|< 2 |
||sin (21*x) otherwise |
\\ /
$$\left(- 3 \left(\begin{cases} 0 & \text{for}\: 21 x \bmod \pi = 0 \\\sin^{2}{\left(21 x \right)} & \text{otherwise} \end{cases}\right)\right) + 14$$
2/pi 21*x\
56*tan |-- + ----|
11*(1 - cos(42*x)) \4 2 /
------------------ + ----------------------
2 2
/ 2/pi 21*x\\
|1 + tan |-- + ----||
\ \4 2 //
$$\frac{11 \cdot \left(- \cos{\left(42 x \right)} + 1\right)}{2} + \frac{56 \tan^{2}{\left(\frac{21 x}{2} + \frac{\pi}{4} \right)}}{\left(\tan^{2}{\left(\frac{21 x}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}}$$
// 0 for 21*x mod pi = 0\
|| |
|| 2/21*x\ |
|| 4*cot |----| |
|| \ 2 / |
14 - 3*|<----------------- otherwise |
|| 2 |
||/ 2/21*x\\ |
|||1 + cot |----|| |
||\ \ 2 // |
\\ /
$$\left(- 3 \left(\begin{cases} 0 & \text{for}\: 21 x \bmod \pi = 0 \\\frac{4 \cot^{2}{\left(\frac{21 x}{2} \right)}}{\left(\cot^{2}{\left(\frac{21 x}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right)\right) + 14$$
2
/ 2/21*x\\ 2/21*x\
14*|1 - tan |----|| 44*tan |----|
\ \ 2 // \ 2 /
-------------------- + -----------------
2 2
/ 2/21*x\\ / 2/21*x\\
|1 + tan |----|| |1 + tan |----||
\ \ 2 // \ \ 2 //
$$\frac{14 \left(- \tan^{2}{\left(\frac{21 x}{2} \right)} + 1\right)^{2}}{\left(\tan^{2}{\left(\frac{21 x}{2} \right)} + 1\right)^{2}} + \frac{44 \tan^{2}{\left(\frac{21 x}{2} \right)}}{\left(\tan^{2}{\left(\frac{21 x}{2} \right)} + 1\right)^{2}}$$
2/21*x\ 2/pi 21*x\
44*tan |----| 56*tan |-- + ----|
\ 2 / \4 2 /
----------------- + ----------------------
2 2
/ 2/21*x\\ / 2/pi 21*x\\
|1 + tan |----|| |1 + tan |-- + ----||
\ \ 2 // \ \4 2 //
$$\frac{56 \tan^{2}{\left(\frac{21 x}{2} + \frac{\pi}{4} \right)}}{\left(\tan^{2}{\left(\frac{21 x}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}} + \frac{44 \tan^{2}{\left(\frac{21 x}{2} \right)}}{\left(\tan^{2}{\left(\frac{21 x}{2} \right)} + 1\right)^{2}}$$
2/21*x\ 2/pi 21*x\
44*cot |----| 56*tan |-- + ----|
\ 2 / \4 2 /
----------------- + ----------------------
2 2
/ 2/21*x\\ / 2/pi 21*x\\
|1 + cot |----|| |1 + tan |-- + ----||
\ \ 2 // \ \4 2 //
$$\frac{44 \cot^{2}{\left(\frac{21 x}{2} \right)}}{\left(\cot^{2}{\left(\frac{21 x}{2} \right)} + 1\right)^{2}} + \frac{56 \tan^{2}{\left(\frac{21 x}{2} + \frac{\pi}{4} \right)}}{\left(\tan^{2}{\left(\frac{21 x}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}}$$
/ 4 8/21*x\ 2 4/21*x\\
2*|7*sin (21*x) + 112*sin |----| + 32*sin (21*x)*sin |----||
\ \ 2 / \ 2 //
------------------------------------------------------------
2
/ 2 4/21*x\\
|sin (21*x) + 4*sin |----||
\ \ 2 //
$$\frac{2 \cdot \left(112 \sin^{8}{\left(\frac{21 x}{2} \right)} + 32 \sin^{4}{\left(\frac{21 x}{2} \right)} \sin^{2}{\left(21 x \right)} + 7 \sin^{4}{\left(21 x \right)}\right)}{\left(4 \sin^{4}{\left(\frac{21 x}{2} \right)} + \sin^{2}{\left(21 x \right)}\right)^{2}}$$
2
/ 1 \
14*|1 - ----------|
| 2/21*x\|
| cot |----||
\ \ 2 // 44
-------------------- + ----------------------------
2 2
/ 1 \ / 1 \ 2/21*x\
|1 + ----------| |1 + ----------| *cot |----|
| 2/21*x\| | 2/21*x\| \ 2 /
| cot |----|| | cot |----||
\ \ 2 // \ \ 2 //
$$\frac{14 \left(1 - \frac{1}{\cot^{2}{\left(\frac{21 x}{2} \right)}}\right)^{2}}{\left(1 + \frac{1}{\cot^{2}{\left(\frac{21 x}{2} \right)}}\right)^{2}} + \frac{44}{\left(1 + \frac{1}{\cot^{2}{\left(\frac{21 x}{2} \right)}}\right)^{2} \cot^{2}{\left(\frac{21 x}{2} \right)}}$$
// 0 for 21*x mod pi = 0\ // 1 for 21*x mod 2*pi = 0\ || | || | 11*|< 2 | + 14*|< 2 | ||sin (21*x) otherwise | ||cos (21*x) otherwise | \\ / \\ /
$$\left(11 \left(\begin{cases} 0 & \text{for}\: 21 x \bmod \pi = 0 \\\sin^{2}{\left(21 x \right)} & \text{otherwise} \end{cases}\right)\right) + \left(14 \left(\begin{cases} 1 & \text{for}\: 21 x \bmod 2 \pi = 0 \\\cos^{2}{\left(21 x \right)} & \text{otherwise} \end{cases}\right)\right)$$
2 2
/ 2/pi 21*x\\ / 2/21*x\\
11*|-1 + tan |-- + ----|| 14*|-1 + cot |----||
\ \4 2 // \ \ 2 //
-------------------------- + ---------------------
2 2
/ 2/pi 21*x\\ / 2/21*x\\
|1 + tan |-- + ----|| |1 + cot |----||
\ \4 2 // \ \ 2 //
$$\frac{11 \left(\tan^{2}{\left(\frac{21 x}{2} + \frac{\pi}{4} \right)} - 1\right)^{2}}{\left(\tan^{2}{\left(\frac{21 x}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}} + \frac{14 \left(\cot^{2}{\left(\frac{21 x}{2} \right)} - 1\right)^{2}}{\left(\cot^{2}{\left(\frac{21 x}{2} \right)} + 1\right)^{2}}$$
2 2
/ 2/pi 21*x\\ / 2/21*x\\
11*|1 - cot |-- + ----|| 14*|1 - tan |----||
\ \4 2 // \ \ 2 //
------------------------- + --------------------
2 2
/ 2/pi 21*x\\ / 2/21*x\\
|1 + cot |-- + ----|| |1 + tan |----||
\ \4 2 // \ \ 2 //
$$\frac{14 \left(- \tan^{2}{\left(\frac{21 x}{2} \right)} + 1\right)^{2}}{\left(\tan^{2}{\left(\frac{21 x}{2} \right)} + 1\right)^{2}} + \frac{11 \left(- \cot^{2}{\left(\frac{21 x}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}}{\left(\cot^{2}{\left(\frac{21 x}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}}$$
// 0 for 21*x mod pi = 0\ // 1 for 21*x mod 2*pi = 0\ || | || | 11*|< 2 | + 14*|< 2/pi \ | ||sin (21*x) otherwise | ||sin |-- + 21*x| otherwise | \\ / \\ \2 / /
$$\left(11 \left(\begin{cases} 0 & \text{for}\: 21 x \bmod \pi = 0 \\\sin^{2}{\left(21 x \right)} & \text{otherwise} \end{cases}\right)\right) + \left(14 \left(\begin{cases} 1 & \text{for}\: 21 x \bmod 2 \pi = 0 \\\sin^{2}{\left(21 x + \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right)\right)$$
// 0 for 21*x mod pi = 0\ // 1 for 21*x mod 2*pi = 0\ || | || | 11*|< 2/ pi\ | + 14*|< 2 | ||cos |21*x - --| otherwise | ||cos (21*x) otherwise | \\ \ 2 / / \\ /
$$\left(11 \left(\begin{cases} 0 & \text{for}\: 21 x \bmod \pi = 0 \\\cos^{2}{\left(21 x - \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right)\right) + \left(14 \left(\begin{cases} 1 & \text{for}\: 21 x \bmod 2 \pi = 0 \\\cos^{2}{\left(21 x \right)} & \text{otherwise} \end{cases}\right)\right)$$
// /pi \ \
// 0 for 21*x mod pi = 0\ || 0 for |-- + 21*x| mod pi = 0|
|| | || \2 / |
11*|< 2 | + 14*|< |
||sin (21*x) otherwise | || 2 |
\\ / ||cos (21*x) otherwise |
\\ /
$$\left(11 \left(\begin{cases} 0 & \text{for}\: 21 x \bmod \pi = 0 \\\sin^{2}{\left(21 x \right)} & \text{otherwise} \end{cases}\right)\right) + \left(14 \left(\begin{cases} 0 & \text{for}\: \left(21 x + \frac{\pi}{2}\right) \bmod \pi = 0 \\\cos^{2}{\left(21 x \right)} & \text{otherwise} \end{cases}\right)\right)$$
// 0 for 21*x mod pi = 0\ // 1 for 21*x mod 2*pi = 0\ || | || | || 1 | || 1 | 11*|<--------------- otherwise | + 14*|<---------- otherwise | || 2/ pi\ | || 2 | ||sec |21*x - --| | ||sec (21*x) | \\ \ 2 / / \\ /
$$\left(11 \left(\begin{cases} 0 & \text{for}\: 21 x \bmod \pi = 0 \\\frac{1}{\sec^{2}{\left(21 x - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) + \left(14 \left(\begin{cases} 1 & \text{for}\: 21 x \bmod 2 \pi = 0 \\\frac{1}{\sec^{2}{\left(21 x \right)}} & \text{otherwise} \end{cases}\right)\right)$$
// 0 for 21*x mod pi = 0\ // 1 for 21*x mod 2*pi = 0\ || | || | || 1 | || 1 | 11*|<---------- otherwise | + 14*|<--------------- otherwise | || 2 | || 2/pi \ | ||csc (21*x) | ||csc |-- - 21*x| | \\ / \\ \2 / /
$$\left(11 \left(\begin{cases} 0 & \text{for}\: 21 x \bmod \pi = 0 \\\frac{1}{\csc^{2}{\left(21 x \right)}} & \text{otherwise} \end{cases}\right)\right) + \left(14 \left(\begin{cases} 1 & \text{for}\: 21 x \bmod 2 \pi = 0 \\\frac{1}{\csc^{2}{\left(- 21 x + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)\right)$$
// / 3*pi\ \ || 1 for |21*x + ----| mod 2*pi = 0| // 1 for 21*x mod 2*pi = 0\ || \ 2 / | || | 11*|< | + 14*|< 2 | || 4/21*x\ 2/21*x\ | ||cos (21*x) otherwise | ||- 4*cos |----| + 4*cos |----| otherwise | \\ / \\ \ 2 / \ 2 / /
$$\left(14 \left(\begin{cases} 1 & \text{for}\: 21 x \bmod 2 \pi = 0 \\\cos^{2}{\left(21 x \right)} & \text{otherwise} \end{cases}\right)\right) + \left(11 \left(\begin{cases} 1 & \text{for}\: \left(21 x + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\- 4 \cos^{4}{\left(\frac{21 x}{2} \right)} + 4 \cos^{2}{\left(\frac{21 x}{2} \right)} & \text{otherwise} \end{cases}\right)\right)$$
2
/ 4/21*x\\
| 4*sin |----||
| \ 2 /|
14*|1 - ------------| 4/21*x\
| 2 | 176*sin |----|
\ sin (21*x) / \ 2 /
---------------------- + ------------------------------
2 2
/ 4/21*x\\ / 4/21*x\\
| 4*sin |----|| | 4*sin |----||
| \ 2 /| | \ 2 /| 2
|1 + ------------| |1 + ------------| *sin (21*x)
| 2 | | 2 |
\ sin (21*x) / \ sin (21*x) /
$$\frac{14 \left(- \frac{4 \sin^{4}{\left(\frac{21 x}{2} \right)}}{\sin^{2}{\left(21 x \right)}} + 1\right)^{2}}{\left(\frac{4 \sin^{4}{\left(\frac{21 x}{2} \right)}}{\sin^{2}{\left(21 x \right)}} + 1\right)^{2}} + \frac{176 \sin^{4}{\left(\frac{21 x}{2} \right)}}{\left(\frac{4 \sin^{4}{\left(\frac{21 x}{2} \right)}}{\sin^{2}{\left(21 x \right)}} + 1\right)^{2} \sin^{2}{\left(21 x \right)}}$$
// 0 for 21*x mod pi = 0\ // 1 for 21*x mod 2*pi = 0\ || | || | ||/ 0 for 21*x mod pi = 0 | ||/ 1 for 21*x mod 2*pi = 0 | 11*|<| | + 14*|<| | ||< 2 otherwise | ||< 2 otherwise | |||sin (21*x) otherwise | |||cos (21*x) otherwise | \\\ / \\\ /
$$\left(11 \left(\begin{cases} 0 & \text{for}\: 21 x \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: 21 x \bmod \pi = 0 \\\sin^{2}{\left(21 x \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)\right) + \left(14 \left(\begin{cases} 1 & \text{for}\: 21 x \bmod 2 \pi = 0 \\\begin{cases} 1 & \text{for}\: 21 x \bmod 2 \pi = 0 \\\cos^{2}{\left(21 x \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)\right)$$
// 0 for 21*x mod pi = 0\ // 1 for 21*x mod 2*pi = 0\ || | || | || 2/21*x\ | || 2 | || 4*cot |----| | ||/ 2/21*x\\ | || \ 2 / | |||-1 + cot |----|| | 11*|<----------------- otherwise | + 14*|<\ \ 2 // | || 2 | ||------------------ otherwise | ||/ 2/21*x\\ | || 2 | |||1 + cot |----|| | ||/ 2/21*x\\ | ||\ \ 2 // | |||1 + cot |----|| | \\ / \\\ \ 2 // /
$$\left(11 \left(\begin{cases} 0 & \text{for}\: 21 x \bmod \pi = 0 \\\frac{4 \cot^{2}{\left(\frac{21 x}{2} \right)}}{\left(\cot^{2}{\left(\frac{21 x}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right)\right) + \left(14 \left(\begin{cases} 1 & \text{for}\: 21 x \bmod 2 \pi = 0 \\\frac{\left(\cot^{2}{\left(\frac{21 x}{2} \right)} - 1\right)^{2}}{\left(\cot^{2}{\left(\frac{21 x}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right)\right)$$
// 0 for 21*x mod pi = 0\ // 1 for 21*x mod 2*pi = 0\ || | || | || 2/21*x\ | || 2 | || 4*tan |----| | ||/ 2/21*x\\ | || \ 2 / | |||1 - tan |----|| | 11*|<----------------- otherwise | + 14*|<\ \ 2 // | || 2 | ||----------------- otherwise | ||/ 2/21*x\\ | || 2 | |||1 + tan |----|| | ||/ 2/21*x\\ | ||\ \ 2 // | |||1 + tan |----|| | \\ / \\\ \ 2 // /
$$\left(11 \left(\begin{cases} 0 & \text{for}\: 21 x \bmod \pi = 0 \\\frac{4 \tan^{2}{\left(\frac{21 x}{2} \right)}}{\left(\tan^{2}{\left(\frac{21 x}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right)\right) + \left(14 \left(\begin{cases} 1 & \text{for}\: 21 x \bmod 2 \pi = 0 \\\frac{\left(- \tan^{2}{\left(\frac{21 x}{2} \right)} + 1\right)^{2}}{\left(\tan^{2}{\left(\frac{21 x}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right)\right)$$
2
/ 2/ pi 21*x\\
| cos |- -- + ----||
| \ 2 2 /|
14*|1 - -----------------|
| 2/21*x\ | 2/ pi 21*x\
| cos |----| | 44*cos |- -- + ----|
\ \ 2 / / \ 2 2 /
--------------------------- + -----------------------------------
2 2
/ 2/ pi 21*x\\ / 2/ pi 21*x\\
| cos |- -- + ----|| | cos |- -- + ----||
| \ 2 2 /| | \ 2 2 /| 2/21*x\
|1 + -----------------| |1 + -----------------| *cos |----|
| 2/21*x\ | | 2/21*x\ | \ 2 /
| cos |----| | | cos |----| |
\ \ 2 / / \ \ 2 / /
$$\frac{14 \left(1 - \frac{\cos^{2}{\left(\frac{21 x}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{21 x}{2} \right)}}\right)^{2}}{\left(1 + \frac{\cos^{2}{\left(\frac{21 x}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{21 x}{2} \right)}}\right)^{2}} + \frac{44 \cos^{2}{\left(\frac{21 x}{2} - \frac{\pi}{2} \right)}}{\left(1 + \frac{\cos^{2}{\left(\frac{21 x}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{21 x}{2} \right)}}\right)^{2} \cos^{2}{\left(\frac{21 x}{2} \right)}}$$
2
/ 2/21*x\ \
| sec |----| |
| \ 2 / |
14*|1 - -----------------|
| 2/ pi 21*x\| 2/21*x\
| sec |- -- + ----|| 44*sec |----|
\ \ 2 2 // \ 2 /
--------------------------- + ------------------------------------------
2 2
/ 2/21*x\ \ / 2/21*x\ \
| sec |----| | | sec |----| |
| \ 2 / | | \ 2 / | 2/ pi 21*x\
|1 + -----------------| |1 + -----------------| *sec |- -- + ----|
| 2/ pi 21*x\| | 2/ pi 21*x\| \ 2 2 /
| sec |- -- + ----|| | sec |- -- + ----||
\ \ 2 2 // \ \ 2 2 //
$$\frac{14 \left(- \frac{\sec^{2}{\left(\frac{21 x}{2} \right)}}{\sec^{2}{\left(\frac{21 x}{2} - \frac{\pi}{2} \right)}} + 1\right)^{2}}{\left(\frac{\sec^{2}{\left(\frac{21 x}{2} \right)}}{\sec^{2}{\left(\frac{21 x}{2} - \frac{\pi}{2} \right)}} + 1\right)^{2}} + \frac{44 \sec^{2}{\left(\frac{21 x}{2} \right)}}{\left(\frac{\sec^{2}{\left(\frac{21 x}{2} \right)}}{\sec^{2}{\left(\frac{21 x}{2} - \frac{\pi}{2} \right)}} + 1\right)^{2} \sec^{2}{\left(\frac{21 x}{2} - \frac{\pi}{2} \right)}}$$
2
/ 2/pi 21*x\\
| csc |-- - ----||
| \2 2 /|
14*|1 - ---------------|
| 2/21*x\ | 2/pi 21*x\
| csc |----| | 44*csc |-- - ----|
\ \ 2 / / \2 2 /
------------------------- + ---------------------------------
2 2
/ 2/pi 21*x\\ / 2/pi 21*x\\
| csc |-- - ----|| | csc |-- - ----||
| \2 2 /| | \2 2 /| 2/21*x\
|1 + ---------------| |1 + ---------------| *csc |----|
| 2/21*x\ | | 2/21*x\ | \ 2 /
| csc |----| | | csc |----| |
\ \ 2 / / \ \ 2 / /
$$\frac{14 \left(1 - \frac{\csc^{2}{\left(- \frac{21 x}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{21 x}{2} \right)}}\right)^{2}}{\left(1 + \frac{\csc^{2}{\left(- \frac{21 x}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{21 x}{2} \right)}}\right)^{2}} + \frac{44 \csc^{2}{\left(- \frac{21 x}{2} + \frac{\pi}{2} \right)}}{\left(1 + \frac{\csc^{2}{\left(- \frac{21 x}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{21 x}{2} \right)}}\right)^{2} \csc^{2}{\left(\frac{21 x}{2} \right)}}$$
// 1 for 21*x mod 2*pi = 0\
|| |
// 0 for 21*x mod pi = 0\ || 2 |
|| | ||/ 1 \ |
|| 4 | |||-1 + ----------| |
||---------------------------- otherwise | ||| 2/21*x\| |
|| 2 | ||| tan |----|| |
11*|
$$\left(11 \left(\begin{cases} 0 & \text{for}\: 21 x \bmod \pi = 0 \\\frac{4}{\left(1 + \frac{1}{\tan^{2}{\left(\frac{21 x}{2} \right)}}\right)^{2} \tan^{2}{\left(\frac{21 x}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) + \left(14 \left(\begin{cases} 1 & \text{for}\: 21 x \bmod 2 \pi = 0 \\\frac{\left(-1 + \frac{1}{\tan^{2}{\left(\frac{21 x}{2} \right)}}\right)^{2}}{\left(1 + \frac{1}{\tan^{2}{\left(\frac{21 x}{2} \right)}}\right)^{2}} & \text{otherwise} \end{cases}\right)\right)$$
// /pi \ \
// 0 for 21*x mod pi = 0\ || 0 for |-- + 21*x| mod pi = 0|
|| | || \2 / |
|| 2/21*x\ | || |
|| 4*cot |----| | || 2/pi 21*x\ |
|| \ 2 / | || 4*cot |-- + ----| |
11*|<----------------- otherwise | + 14*|< \4 2 / |
|| 2 | ||---------------------- otherwise |
||/ 2/21*x\\ | || 2 |
|||1 + cot |----|| | ||/ 2/pi 21*x\\ |
||\ \ 2 // | |||1 + cot |-- + ----|| |
\\ / ||\ \4 2 // |
\\ /
$$\left(11 \left(\begin{cases} 0 & \text{for}\: 21 x \bmod \pi = 0 \\\frac{4 \cot^{2}{\left(\frac{21 x}{2} \right)}}{\left(\cot^{2}{\left(\frac{21 x}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right)\right) + \left(14 \left(\begin{cases} 0 & \text{for}\: \left(21 x + \frac{\pi}{2}\right) \bmod \pi = 0 \\\frac{4 \cot^{2}{\left(\frac{21 x}{2} + \frac{\pi}{4} \right)}}{\left(\cot^{2}{\left(\frac{21 x}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right)\right)$$
// / 3*pi\ \ || 1 for |21*x + ----| mod 2*pi = 0| // 1 for 21*x mod 2*pi = 0\ || \ 2 / | || | || | || 2 | || 2 | ||/ 2/21*x\\ | ||/ 2/pi 21*x\\ | |||-1 + cot |----|| | 11*|<|-1 + tan |-- + ----|| | + 14*|<\ \ 2 // | ||\ \4 2 // | ||------------------ otherwise | ||----------------------- otherwise | || 2 | || 2 | ||/ 2/21*x\\ | || / 2/pi 21*x\\ | |||1 + cot |----|| | || |1 + tan |-- + ----|| | \\\ \ 2 // / \\ \ \4 2 // /
$$\left(14 \left(\begin{cases} 1 & \text{for}\: 21 x \bmod 2 \pi = 0 \\\frac{\left(\cot^{2}{\left(\frac{21 x}{2} \right)} - 1\right)^{2}}{\left(\cot^{2}{\left(\frac{21 x}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right)\right) + \left(11 \left(\begin{cases} 1 & \text{for}\: \left(21 x + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\frac{\left(\tan^{2}{\left(\frac{21 x}{2} + \frac{\pi}{4} \right)} - 1\right)^{2}}{\left(\tan^{2}{\left(\frac{21 x}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right)\right)$$
// 0 for 21*x mod pi = 0\
|| | // 1 for 21*x mod 2*pi = 0\
|| 2 | || |
|| sin (21*x) | || 2 |
||------------------------------ otherwise | ||/ 2 4/21*x\\ |
|| 2 | |||sin (21*x) - 4*sin |----|| |
11*|
$$\left(11 \left(\begin{cases} 0 & \text{for}\: 21 x \bmod \pi = 0 \\\frac{\sin^{2}{\left(21 x \right)}}{\left(1 + \frac{\sin^{2}{\left(21 x \right)}}{4 \sin^{4}{\left(\frac{21 x}{2} \right)}}\right)^{2} \sin^{4}{\left(\frac{21 x}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) + \left(14 \left(\begin{cases} 1 & \text{for}\: 21 x \bmod 2 \pi = 0 \\\frac{\left(- 4 \sin^{4}{\left(\frac{21 x}{2} \right)} + \sin^{2}{\left(21 x \right)}\right)^{2}}{\left(4 \sin^{4}{\left(\frac{21 x}{2} \right)} + \sin^{2}{\left(21 x \right)}\right)^{2}} & \text{otherwise} \end{cases}\right)\right)$$
// 1 for 21*x mod 2*pi = 0\
|| |
// 0 for 21*x mod pi = 0\ || 2 |
|| | ||/ 2 \ |
|| 2 | ||| sin (21*x) | |
|| sin (21*x) | |||-1 + ------------| |
||------------------------------ otherwise | ||| 4/21*x\| |
|| 2 | ||| 4*sin |----|| |
11*|
$$\left(11 \left(\begin{cases} 0 & \text{for}\: 21 x \bmod \pi = 0 \\\frac{\sin^{2}{\left(21 x \right)}}{\left(1 + \frac{\sin^{2}{\left(21 x \right)}}{4 \sin^{4}{\left(\frac{21 x}{2} \right)}}\right)^{2} \sin^{4}{\left(\frac{21 x}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) + \left(14 \left(\begin{cases} 1 & \text{for}\: 21 x \bmod 2 \pi = 0 \\\frac{\left(-1 + \frac{\sin^{2}{\left(21 x \right)}}{4 \sin^{4}{\left(\frac{21 x}{2} \right)}}\right)^{2}}{\left(1 + \frac{\sin^{2}{\left(21 x \right)}}{4 \sin^{4}{\left(\frac{21 x}{2} \right)}}\right)^{2}} & \text{otherwise} \end{cases}\right)\right)$$
// 0 for 21*x mod pi = 0\ // 1 for 21*x mod 2*pi = 0\ || | || | ||/ 0 for 21*x mod pi = 0 | ||/ 1 for 21*x mod 2*pi = 0 | ||| | ||| | ||| 2/21*x\ | ||| 2 | ||| 4*cot |----| | |||/ 2/21*x\\ | 11*|<| \ 2 / | + 14*|<||-1 + cot |----|| | ||<----------------- otherwise otherwise | ||<\ \ 2 // otherwise | ||| 2 | |||------------------ otherwise | |||/ 2/21*x\\ | ||| 2 | ||||1 + cot |----|| | |||/ 2/21*x\\ | |||\ \ 2 // | ||||1 + cot |----|| | \\\ / \\\\ \ 2 // /
$$\left(11 \left(\begin{cases} 0 & \text{for}\: 21 x \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: 21 x \bmod \pi = 0 \\\frac{4 \cot^{2}{\left(\frac{21 x}{2} \right)}}{\left(\cot^{2}{\left(\frac{21 x}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)\right) + \left(14 \left(\begin{cases} 1 & \text{for}\: 21 x \bmod 2 \pi = 0 \\\begin{cases} 1 & \text{for}\: 21 x \bmod 2 \pi = 0 \\\frac{\left(\cot^{2}{\left(\frac{21 x}{2} \right)} - 1\right)^{2}}{\left(\cot^{2}{\left(\frac{21 x}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)\right)$$
// 1 for 21*x mod 2*pi = 0\
|| |
// 0 for 21*x mod pi = 0\ || 2 |
|| | ||/ 2/21*x\ \ |
|| 2/21*x\ | ||| cos |----| | |
|| 4*cos |----| | ||| \ 2 / | |
|| \ 2 / | |||-1 + -----------------| |
||------------------------------------------ otherwise | ||| 2/ pi 21*x\| |
|| 2 | ||| cos |- -- + ----|| |
11*|
$$\left(11 \left(\begin{cases} 0 & \text{for}\: 21 x \bmod \pi = 0 \\\frac{4 \cos^{2}{\left(\frac{21 x}{2} \right)}}{\left(\frac{\cos^{2}{\left(\frac{21 x}{2} \right)}}{\cos^{2}{\left(\frac{21 x}{2} - \frac{\pi}{2} \right)}} + 1\right)^{2} \cos^{2}{\left(\frac{21 x}{2} - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) + \left(14 \left(\begin{cases} 1 & \text{for}\: 21 x \bmod 2 \pi = 0 \\\frac{\left(\frac{\cos^{2}{\left(\frac{21 x}{2} \right)}}{\cos^{2}{\left(\frac{21 x}{2} - \frac{\pi}{2} \right)}} - 1\right)^{2}}{\left(\frac{\cos^{2}{\left(\frac{21 x}{2} \right)}}{\cos^{2}{\left(\frac{21 x}{2} - \frac{\pi}{2} \right)}} + 1\right)^{2}} & \text{otherwise} \end{cases}\right)\right)$$
// 1 for 21*x mod 2*pi = 0\
|| |
// 0 for 21*x mod pi = 0\ || 2 |
|| | ||/ 2/ pi 21*x\\ |
|| 2/ pi 21*x\ | ||| sec |- -- + ----|| |
|| 4*sec |- -- + ----| | ||| \ 2 2 /| |
|| \ 2 2 / | |||-1 + -----------------| |
||----------------------------------- otherwise | ||| 2/21*x\ | |
|| 2 | ||| sec |----| | |
11*|
$$\left(11 \left(\begin{cases} 0 & \text{for}\: 21 x \bmod \pi = 0 \\\frac{4 \sec^{2}{\left(\frac{21 x}{2} - \frac{\pi}{2} \right)}}{\left(1 + \frac{\sec^{2}{\left(\frac{21 x}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{21 x}{2} \right)}}\right)^{2} \sec^{2}{\left(\frac{21 x}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) + \left(14 \left(\begin{cases} 1 & \text{for}\: 21 x \bmod 2 \pi = 0 \\\frac{\left(-1 + \frac{\sec^{2}{\left(\frac{21 x}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{21 x}{2} \right)}}\right)^{2}}{\left(1 + \frac{\sec^{2}{\left(\frac{21 x}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{21 x}{2} \right)}}\right)^{2}} & \text{otherwise} \end{cases}\right)\right)$$
// 1 for 21*x mod 2*pi = 0\
|| |
// 0 for 21*x mod pi = 0\ || 2 |
|| | ||/ 2/21*x\ \ |
|| 2/21*x\ | ||| csc |----| | |
|| 4*csc |----| | ||| \ 2 / | |
|| \ 2 / | |||-1 + ---------------| |
||-------------------------------------- otherwise | ||| 2/pi 21*x\| |
|| 2 | ||| csc |-- - ----|| |
11*|
$$\left(11 \left(\begin{cases} 0 & \text{for}\: 21 x \bmod \pi = 0 \\\frac{4 \csc^{2}{\left(\frac{21 x}{2} \right)}}{\left(\frac{\csc^{2}{\left(\frac{21 x}{2} \right)}}{\csc^{2}{\left(- \frac{21 x}{2} + \frac{\pi}{2} \right)}} + 1\right)^{2} \csc^{2}{\left(- \frac{21 x}{2} + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) + \left(14 \left(\begin{cases} 1 & \text{for}\: 21 x \bmod 2 \pi = 0 \\\frac{\left(\frac{\csc^{2}{\left(\frac{21 x}{2} \right)}}{\csc^{2}{\left(- \frac{21 x}{2} + \frac{\pi}{2} \right)}} - 1\right)^{2}}{\left(\frac{\csc^{2}{\left(\frac{21 x}{2} \right)}}{\csc^{2}{\left(- \frac{21 x}{2} + \frac{\pi}{2} \right)}} + 1\right)^{2}} & \text{otherwise} \end{cases}\right)\right)$$
11*Piecewise((0, Mod(21*x = pi, 0)), (4*csc(21*x/2)^2/((1 + csc(21*x/2)^2/csc(pi/2 - 21*x/2)^2)^2*csc(pi/2 - 21*x/2)^2), True)) + 14*Piecewise((1, Mod(21*x = 2*pi, 0)), ((-1 + csc(21*x/2)^2/csc(pi/2 - 21*x/2)^2)^2/(1 + csc(21*x/2)^2/csc(pi/2 - 21*x/2)^2)^2, True))
Степени
[src]
2 2 / -21*I*x 21*I*x\ / -21*I*x 21*I*x\ |e e | 11*\- e + e / 14*|-------- + -------| - -------------------------- \ 2 2 / 4
$$14 \left(\frac{e^{21 i x}}{2} + \frac{e^{- 21 i x}}{2}\right)^{2} - \frac{11 \left(e^{21 i x} - e^{- 21 i x}\right)^{2}}{4}$$
14*(exp(-21*i*x)/2 + exp(21*i*x)/2)^2 - 11*(-exp(-21*i*x) + exp(21*i*x))^2/4