Господин Экзамен
sin(6*x)-cos(6*x) если x=2
Решение
Вы ввели
[src]
sin(6*x) - cos(6*x)
$$\sin{\left(6 x \right)} - \cos{\left(6 x \right)}$$
sin(6*x) - cos(6*x)
Общее упрощение
[src]
___ / pi\
-\/ 2 *cos|6*x + --|
\ 4 /
$$- \sqrt{2} \cos{\left(6 x + \frac{\pi}{4} \right)}$$
-sqrt(2)*cos(6*x + pi/4)
Подстановка условия
[src]
sin(6*x) - cos(6*x) при x = 2
подставляем
sin(6*x) - cos(6*x)
$$\sin{\left(6 x \right)} - \cos{\left(6 x \right)}$$
___ / pi\
-\/ 2 *cos|6*x + --|
\ 4 /
$$- \sqrt{2} \cos{\left(6 x + \frac{\pi}{4} \right)}$$
переменные
x = 2
$$x = 2$$
___ / pi\
-\/ 2 *cos|6*(2) + --|
\ 4 /
$$- \sqrt{2} \cos{\left(6 (2) + \frac{\pi}{4} \right)}$$
___ / pi\
-\/ 2 *cos|6*2 + --|
\ 4 /
$$- \sqrt{2} \cos{\left(\frac{\pi}{4} + 6 \cdot 2 \right)}$$
___ / pi\
-\/ 2 *cos|12 + --|
\ 4 /
$$- \sqrt{2} \cos{\left(\frac{\pi}{4} + 12 \right)}$$
-sqrt(2)*cos(12 + pi/4)
Тригонометрическая часть
[src]
___ / pi\
-\/ 2 *cos|6*x + --|
\ 4 /
$$- \sqrt{2} \cos{\left(6 x + \frac{\pi}{4} \right)}$$
___ / 3*pi\
-\/ 2 *sin|6*x + ----|
\ 4 /
$$- \sqrt{2} \sin{\left(6 x + \frac{3 \pi}{4} \right)}$$
___ -\/ 2 ------------- / pi\ sec|6*x + --| \ 4 /
$$- \frac{\sqrt{2}}{\sec{\left(6 x + \frac{\pi}{4} \right)}}$$
___ -\/ 2 -------------- / pi\ csc|-6*x + --| \ 4 /
$$- \frac{\sqrt{2}}{\csc{\left(- 6 x + \frac{\pi}{4} \right)}}$$
2 1 - 2*cos (3*x) + sin(6*x)
$$- 2 \cos^{2}{\left(3 x \right)} + \sin{\left(6 x \right)} + 1$$
/pi \
- sin|-- + 6*x| + sin(6*x)
\2 /
$$\sin{\left(6 x \right)} - \sin{\left(6 x + \frac{\pi}{2} \right)}$$
/ pi\
-cos(6*x) + cos|6*x - --|
\ 2 /
$$- \cos{\left(6 x \right)} + \cos{\left(6 x - \frac{\pi}{2} \right)}$$
1 1 -------- - -------- csc(6*x) sec(6*x)
$$- \frac{1}{\sec{\left(6 x \right)}} + \frac{1}{\csc{\left(6 x \right)}}$$
1 1 ------------- - -------- / pi\ sec(6*x) sec|6*x - --| \ 2 /
$$\frac{1}{\sec{\left(6 x - \frac{\pi}{2} \right)}} - \frac{1}{\sec{\left(6 x \right)}}$$
1 1 ------------- - -------- /pi \ sec(6*x) sec|-- - 6*x| \2 /
$$\frac{1}{\sec{\left(- 6 x + \frac{\pi}{2} \right)}} - \frac{1}{\sec{\left(6 x \right)}}$$
1 1
-------- - -------------
csc(6*x) /pi \
csc|-- - 6*x|
\2 /
$$- \frac{1}{\csc{\left(- 6 x + \frac{\pi}{2} \right)}} + \frac{1}{\csc{\left(6 x \right)}}$$
1 1
------------- - -------------
csc(pi - 6*x) /pi \
csc|-- - 6*x|
\2 /
$$- \frac{1}{\csc{\left(- 6 x + \frac{\pi}{2} \right)}} + \frac{1}{\csc{\left(- 6 x + \pi \right)}}$$
2 2 sin (3*x) - cos (3*x) + 2*cos(3*x)*sin(3*x)
$$\sin^{2}{\left(3 x \right)} + 2 \sin{\left(3 x \right)} \cos{\left(3 x \right)} - \cos^{2}{\left(3 x \right)}$$
___ / 2/ pi\\
-\/ 2 *|1 - tan |3*x + --||
\ \ 8 //
----------------------------
2/ pi\
1 + tan |3*x + --|
\ 8 /
$$- \frac{\sqrt{2} \cdot \left(- \tan^{2}{\left(3 x + \frac{\pi}{8} \right)} + 1\right)}{\tan^{2}{\left(3 x + \frac{\pi}{8} \right)} + 1}$$
/ 2/ pi\\
|1 - cot |3*x + --||*(1 + sin(6*x))
\ \ 4 //
-cos(6*x) + -----------------------------------
2
$$\frac{\left(- \cot^{2}{\left(3 x + \frac{\pi}{4} \right)} + 1\right) \left(\sin{\left(6 x \right)} + 1\right)}{2} - \cos{\left(6 x \right)}$$
2
1 - tan (3*x) 2*tan(3*x)
- ------------- + -------------
2 2
1 + tan (3*x) 1 + tan (3*x)
$$- \frac{- \tan^{2}{\left(3 x \right)} + 1}{\tan^{2}{\left(3 x \right)} + 1} + \frac{2 \tan{\left(3 x \right)}}{\tan^{2}{\left(3 x \right)} + 1}$$
/ pi\
2*tan|3*x + --|
\ 4 / 2*cot(3*x)
- ------------------ + -------------
2/ pi\ 2
1 + tan |3*x + --| 1 + cot (3*x)
\ 4 /
$$\frac{2 \cot{\left(3 x \right)}}{\cot^{2}{\left(3 x \right)} + 1} - \frac{2 \tan{\left(3 x + \frac{\pi}{4} \right)}}{\tan^{2}{\left(3 x + \frac{\pi}{4} \right)} + 1}$$
/ pi\
2*tan|3*x + --|
\ 4 / 2*tan(3*x)
- ------------------ + -------------
2/ pi\ 2
1 + tan |3*x + --| 1 + tan (3*x)
\ 4 /
$$- \frac{2 \tan{\left(3 x + \frac{\pi}{4} \right)}}{\tan^{2}{\left(3 x + \frac{\pi}{4} \right)} + 1} + \frac{2 \tan{\left(3 x \right)}}{\tan^{2}{\left(3 x \right)} + 1}$$
2/ pi\
-1 + tan |3*x + --| 2
\ 4 / -1 + cot (3*x)
------------------- - --------------
2/ pi\ 2
1 + tan |3*x + --| 1 + cot (3*x)
\ 4 /
$$\frac{\tan^{2}{\left(3 x + \frac{\pi}{4} \right)} - 1}{\tan^{2}{\left(3 x + \frac{\pi}{4} \right)} + 1} - \frac{\cot^{2}{\left(3 x \right)} - 1}{\cot^{2}{\left(3 x \right)} + 1}$$
2/ pi\
1 - cot |3*x + --| 2
\ 4 / 1 - tan (3*x)
------------------ - -------------
2/ pi\ 2
1 + cot |3*x + --| 1 + tan (3*x)
\ 4 /
$$- \frac{- \tan^{2}{\left(3 x \right)} + 1}{\tan^{2}{\left(3 x \right)} + 1} + \frac{- \cot^{2}{\left(3 x + \frac{\pi}{4} \right)} + 1}{\cot^{2}{\left(3 x + \frac{\pi}{4} \right)} + 1}$$
1
1 - ---------
2
cot (3*x) 2
- ------------- + ------------------------
1 / 1 \
1 + --------- |1 + ---------|*cot(3*x)
2 | 2 |
cot (3*x) \ cot (3*x)/
$$- \frac{1 - \frac{1}{\cot^{2}{\left(3 x \right)}}}{1 + \frac{1}{\cot^{2}{\left(3 x \right)}}} + \frac{2}{\left(1 + \frac{1}{\cot^{2}{\left(3 x \right)}}\right) \cot{\left(3 x \right)}}$$
// 1 for 3*x mod pi = 0\ // 0 for 6*x mod pi = 0\ - |< | + |< | \\cos(6*x) otherwise / \\sin(6*x) otherwise /
$$\left(\begin{cases} 0 & \text{for}\: 6 x \bmod \pi = 0 \\\sin{\left(6 x \right)} & \text{otherwise} \end{cases}\right) - \left(\begin{cases} 1 & \text{for}\: 3 x \bmod \pi = 0 \\\cos{\left(6 x \right)} & \text{otherwise} \end{cases}\right)$$
// / pi\ \
|| 1 for |6*x + --| mod 2*pi = 0|
|| \ 4 / |
|| |
___ || 2/ pi\ |
-\/ 2 *|<-1 + cot |3*x + --| |
|| \ 8 / |
||------------------- otherwise |
|| 2/ pi\ |
|| 1 + cot |3*x + --| |
\\ \ 8 / /
$$- \sqrt{2} \left(\begin{cases} 1 & \text{for}\: \left(6 x + \frac{\pi}{4}\right) \bmod 2 \pi = 0 \\\frac{\cot^{2}{\left(3 x + \frac{\pi}{8} \right)} - 1}{\cot^{2}{\left(3 x + \frac{\pi}{8} \right)} + 1} & \text{otherwise} \end{cases}\right)$$
// / pi\ \
|| 1 for |6*x + --| mod 2*pi = 0|
___ || \ 4 / |
-\/ 2 *|< |
|| 2/ pi\ / 2/ pi\\ |
||sin |3*x + --|*|-1 + cot |3*x + --|| otherwise |
\\ \ 8 / \ \ 8 // /
$$- \sqrt{2} \left(\begin{cases} 1 & \text{for}\: \left(6 x + \frac{\pi}{4}\right) \bmod 2 \pi = 0 \\\left(\cot^{2}{\left(3 x + \frac{\pi}{8} \right)} - 1\right) \sin^{2}{\left(3 x + \frac{\pi}{8} \right)} & \text{otherwise} \end{cases}\right)$$
// 1 for 3*x mod pi = 0\ || | // 0 for 6*x mod pi = 0\ - |< /pi \ | + |< | ||sin|-- + 6*x| otherwise | \\sin(6*x) otherwise / \\ \2 / /
$$\left(\begin{cases} 0 & \text{for}\: 6 x \bmod \pi = 0 \\\sin{\left(6 x \right)} & \text{otherwise} \end{cases}\right) - \left(\begin{cases} 1 & \text{for}\: 3 x \bmod \pi = 0 \\\sin{\left(6 x + \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right)$$
// 0 for 6*x mod pi = 0\
// 1 for 3*x mod pi = 0\ || |
- |< | + |< / pi\ |
\\cos(6*x) otherwise / ||cos|6*x - --| otherwise |
\\ \ 2 / /
$$\left(\begin{cases} 0 & \text{for}\: 6 x \bmod \pi = 0 \\\cos{\left(6 x - \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right) - \left(\begin{cases} 1 & \text{for}\: 3 x \bmod \pi = 0 \\\cos{\left(6 x \right)} & \text{otherwise} \end{cases}\right)$$
// / 3*pi\ \
// 1 for 3*x mod pi = 0\ || 1 for |6*x + ----| mod 2*pi = 0|
- |< | + |< \ 2 / |
\\cos(6*x) otherwise / || |
\\sin(6*x) otherwise /
$$\left(- \begin{cases} 1 & \text{for}\: 3 x \bmod \pi = 0 \\\cos{\left(6 x \right)} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: \left(6 x + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\sin{\left(6 x \right)} & \text{otherwise} \end{cases}\right)$$
// 0 for 6*x mod pi = 0\
// 1 for 3*x mod pi = 0\ || |
|| | || 1 |
- |< 1 | + |<------------- otherwise |
||-------- otherwise | || / pi\ |
\\sec(6*x) / ||sec|6*x - --| |
\\ \ 2 / /
$$\left(\begin{cases} 0 & \text{for}\: 6 x \bmod \pi = 0 \\\frac{1}{\sec{\left(6 x - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) - \left(\begin{cases} 1 & \text{for}\: 3 x \bmod \pi = 0 \\\frac{1}{\sec{\left(6 x \right)}} & \text{otherwise} \end{cases}\right)$$
// 1 for 3*x mod pi = 0\ || | // 0 for 6*x mod pi = 0\ || 1 | || | - |<------------- otherwise | + |< 1 | || /pi \ | ||-------- otherwise | ||csc|-- - 6*x| | \\csc(6*x) / \\ \2 / /
$$\left(\begin{cases} 0 & \text{for}\: 6 x \bmod \pi = 0 \\\frac{1}{\csc{\left(6 x \right)}} & \text{otherwise} \end{cases}\right) - \left(\begin{cases} 1 & \text{for}\: 3 x \bmod \pi = 0 \\\frac{1}{\csc{\left(- 6 x + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)$$
4
4*sin (3*x)
1 - -----------
2 2
sin (6*x) 4*sin (3*x)
- --------------- + --------------------------
4 / 4 \
4*sin (3*x) | 4*sin (3*x)|
1 + ----------- |1 + -----------|*sin(6*x)
2 | 2 |
sin (6*x) \ sin (6*x) /
$$- \frac{- \frac{4 \sin^{4}{\left(3 x \right)}}{\sin^{2}{\left(6 x \right)}} + 1}{\frac{4 \sin^{4}{\left(3 x \right)}}{\sin^{2}{\left(6 x \right)}} + 1} + \frac{4 \sin^{2}{\left(3 x \right)}}{\left(\frac{4 \sin^{4}{\left(3 x \right)}}{\sin^{2}{\left(6 x \right)}} + 1\right) \sin{\left(6 x \right)}}$$
// /pi \ \ || 0 for |-- + 6*x| mod pi = 0| || \2 / | // 0 for 6*x mod pi = 0\ - |< | + |< | || / pi\ | \\sin(6*x) otherwise / ||(1 + sin(6*x))*cot|3*x + --| otherwise | \\ \ 4 / /
$$\left(\begin{cases} 0 & \text{for}\: 6 x \bmod \pi = 0 \\\sin{\left(6 x \right)} & \text{otherwise} \end{cases}\right) - \left(\begin{cases} 0 & \text{for}\: \left(6 x + \frac{\pi}{2}\right) \bmod \pi = 0 \\\left(\sin{\left(6 x \right)} + 1\right) \cot{\left(3 x + \frac{\pi}{4} \right)} & \text{otherwise} \end{cases}\right)$$
// 1 for 3*x mod pi = 0\ // 0 for 6*x mod pi = 0\ || | || | || 2 | || 2*cot(3*x) | - |<-1 + cot (3*x) | + |<------------- otherwise | ||-------------- otherwise | || 2 | || 2 | ||1 + cot (3*x) | \\1 + cot (3*x) / \\ /
$$\left(\begin{cases} 0 & \text{for}\: 6 x \bmod \pi = 0 \\\frac{2 \cot{\left(3 x \right)}}{\cot^{2}{\left(3 x \right)} + 1} & \text{otherwise} \end{cases}\right) - \left(\begin{cases} 1 & \text{for}\: 3 x \bmod \pi = 0 \\\frac{\cot^{2}{\left(3 x \right)} - 1}{\cot^{2}{\left(3 x \right)} + 1} & \text{otherwise} \end{cases}\right)$$
// 1 for 3*x mod pi = 0\ // 0 for 6*x mod pi = 0\ || | || | || 2 | || 2*tan(3*x) | - |<1 - tan (3*x) | + |<------------- otherwise | ||------------- otherwise | || 2 | || 2 | ||1 + tan (3*x) | \\1 + tan (3*x) / \\ /
$$\left(\begin{cases} 0 & \text{for}\: 6 x \bmod \pi = 0 \\\frac{2 \tan{\left(3 x \right)}}{\tan^{2}{\left(3 x \right)} + 1} & \text{otherwise} \end{cases}\right) - \left(\begin{cases} 1 & \text{for}\: 3 x \bmod \pi = 0 \\\frac{- \tan^{2}{\left(3 x \right)} + 1}{\tan^{2}{\left(3 x \right)} + 1} & \text{otherwise} \end{cases}\right)$$
2/ pi\
cos |3*x - --|
\ 2 /
1 - -------------- / pi\
2 2*cos|3*x - --|
cos (3*x) \ 2 /
- ------------------ + -----------------------------
2/ pi\ / 2/ pi\\
cos |3*x - --| | cos |3*x - --||
\ 2 / | \ 2 /|
1 + -------------- |1 + --------------|*cos(3*x)
2 | 2 |
cos (3*x) \ cos (3*x) /
$$- \frac{1 - \frac{\cos^{2}{\left(3 x - \frac{\pi}{2} \right)}}{\cos^{2}{\left(3 x \right)}}}{1 + \frac{\cos^{2}{\left(3 x - \frac{\pi}{2} \right)}}{\cos^{2}{\left(3 x \right)}}} + \frac{2 \cos{\left(3 x - \frac{\pi}{2} \right)}}{\left(1 + \frac{\cos^{2}{\left(3 x - \frac{\pi}{2} \right)}}{\cos^{2}{\left(3 x \right)}}\right) \cos{\left(3 x \right)}}$$
2
sec (3*x)
1 - --------------
2/ pi\
sec |3*x - --|
\ 2 / 2*sec(3*x)
- ------------------ + ----------------------------------
2 / 2 \
sec (3*x) | sec (3*x) | / pi\
1 + -------------- |1 + --------------|*sec|3*x - --|
2/ pi\ | 2/ pi\| \ 2 /
sec |3*x - --| | sec |3*x - --||
\ 2 / \ \ 2 //
$$- \frac{- \frac{\sec^{2}{\left(3 x \right)}}{\sec^{2}{\left(3 x - \frac{\pi}{2} \right)}} + 1}{\frac{\sec^{2}{\left(3 x \right)}}{\sec^{2}{\left(3 x - \frac{\pi}{2} \right)}} + 1} + \frac{2 \sec{\left(3 x \right)}}{\left(\frac{\sec^{2}{\left(3 x \right)}}{\sec^{2}{\left(3 x - \frac{\pi}{2} \right)}} + 1\right) \sec{\left(3 x - \frac{\pi}{2} \right)}}$$
2/pi \
csc |-- - 3*x|
\2 /
1 - -------------- /pi \
2 2*csc|-- - 3*x|
csc (3*x) \2 /
- ------------------ + -----------------------------
2/pi \ / 2/pi \\
csc |-- - 3*x| | csc |-- - 3*x||
\2 / | \2 /|
1 + -------------- |1 + --------------|*csc(3*x)
2 | 2 |
csc (3*x) \ csc (3*x) /
$$- \frac{1 - \frac{\csc^{2}{\left(- 3 x + \frac{\pi}{2} \right)}}{\csc^{2}{\left(3 x \right)}}}{1 + \frac{\csc^{2}{\left(- 3 x + \frac{\pi}{2} \right)}}{\csc^{2}{\left(3 x \right)}}} + \frac{2 \csc{\left(- 3 x + \frac{\pi}{2} \right)}}{\left(1 + \frac{\csc^{2}{\left(- 3 x + \frac{\pi}{2} \right)}}{\csc^{2}{\left(3 x \right)}}\right) \csc{\left(3 x \right)}}$$
// 1 for 3*x mod pi = 0\
|| | // 0 for 6*x mod pi = 0\
|| 1 | || |
||-1 + --------- | || 2 |
|| 2 | ||------------------------ otherwise |
- |< tan (3*x) | + |
$$\left(\begin{cases} 0 & \text{for}\: 6 x \bmod \pi = 0 \\\frac{2}{\left(1 + \frac{1}{\tan^{2}{\left(3 x \right)}}\right) \tan{\left(3 x \right)}} & \text{otherwise} \end{cases}\right) - \left(\begin{cases} 1 & \text{for}\: 3 x \bmod \pi = 0 \\\frac{-1 + \frac{1}{\tan^{2}{\left(3 x \right)}}}{1 + \frac{1}{\tan^{2}{\left(3 x \right)}}} & \text{otherwise} \end{cases}\right)$$
// /pi \ \ || 0 for |-- + 6*x| mod pi = 0| || \2 / | // 0 for 6*x mod pi = 0\ || | || | || / pi\ | || 2*cot(3*x) | - |< 2*cot|3*x + --| | + |<------------- otherwise | || \ 4 / | || 2 | ||------------------ otherwise | ||1 + cot (3*x) | || 2/ pi\ | \\ / ||1 + cot |3*x + --| | \\ \ 4 / /
$$\left(\begin{cases} 0 & \text{for}\: 6 x \bmod \pi = 0 \\\frac{2 \cot{\left(3 x \right)}}{\cot^{2}{\left(3 x \right)} + 1} & \text{otherwise} \end{cases}\right) - \left(\begin{cases} 0 & \text{for}\: \left(6 x + \frac{\pi}{2}\right) \bmod \pi = 0 \\\frac{2 \cot{\left(3 x + \frac{\pi}{4} \right)}}{\cot^{2}{\left(3 x + \frac{\pi}{4} \right)} + 1} & \text{otherwise} \end{cases}\right)$$
// / 3*pi\ \
|| 1 for |6*x + ----| mod 2*pi = 0|
// 1 for 3*x mod pi = 0\ || \ 2 / |
|| | || |
|| 2 | || 2/ pi\ |
- |<-1 + cot (3*x) | + |<-1 + tan |3*x + --| |
||-------------- otherwise | || \ 4 / |
|| 2 | ||------------------- otherwise |
\\1 + cot (3*x) / || 2/ pi\ |
|| 1 + tan |3*x + --| |
\\ \ 4 / /
$$\left(- \begin{cases} 1 & \text{for}\: 3 x \bmod \pi = 0 \\\frac{\cot^{2}{\left(3 x \right)} - 1}{\cot^{2}{\left(3 x \right)} + 1} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: \left(6 x + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\frac{\tan^{2}{\left(3 x + \frac{\pi}{4} \right)} - 1}{\tan^{2}{\left(3 x + \frac{\pi}{4} \right)} + 1} & \text{otherwise} \end{cases}\right)$$
// 1 for 3*x mod pi = 0\ // 0 for 6*x mod pi = 0\
|| | || |
- |
$$\left(\begin{cases} 0 & \text{for}\: 6 x \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: 6 x \bmod \pi = 0 \\\sin{\left(6 x \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) - \left(\begin{cases} 1 & \text{for}\: 3 x \bmod \pi = 0 \\\begin{cases} 1 & \text{for}\: 3 x \bmod \pi = 0 \\\cos{\left(6 x \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)$$
// 1 for 3*x mod pi = 0\
|| |
|| 2 | // 0 for 6*x mod pi = 0\
|| sin (6*x) | || |
||-1 + ----------- | || sin(6*x) |
|| 4 | ||--------------------------- otherwise |
- |< 4*sin (3*x) | + |
$$\left(\begin{cases} 0 & \text{for}\: 6 x \bmod \pi = 0 \\\frac{\sin{\left(6 x \right)}}{\left(1 + \frac{\sin^{2}{\left(6 x \right)}}{4 \sin^{4}{\left(3 x \right)}}\right) \sin^{2}{\left(3 x \right)}} & \text{otherwise} \end{cases}\right) - \left(\begin{cases} 1 & \text{for}\: 3 x \bmod \pi = 0 \\\frac{-1 + \frac{\sin^{2}{\left(6 x \right)}}{4 \sin^{4}{\left(3 x \right)}}}{1 + \frac{\sin^{2}{\left(6 x \right)}}{4 \sin^{4}{\left(3 x \right)}}} & \text{otherwise} \end{cases}\right)$$
// 1 for 3*x mod pi = 0\ // 0 for 6*x mod pi = 0\ || | || | ||/ 1 for 3*x mod pi = 0 | ||/ 0 for 6*x mod pi = 0 | ||| | ||| | - |<| 2 | + |<| 2*cot(3*x) | ||<-1 + cot (3*x) otherwise | ||<------------- otherwise otherwise | |||-------------- otherwise | ||| 2 | ||| 2 | |||1 + cot (3*x) | \\\1 + cot (3*x) / \\\ /
$$\left(\begin{cases} 0 & \text{for}\: 6 x \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: 6 x \bmod \pi = 0 \\\frac{2 \cot{\left(3 x \right)}}{\cot^{2}{\left(3 x \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) - \left(\begin{cases} 1 & \text{for}\: 3 x \bmod \pi = 0 \\\begin{cases} 1 & \text{for}\: 3 x \bmod \pi = 0 \\\frac{\cot^{2}{\left(3 x \right)} - 1}{\cot^{2}{\left(3 x \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)$$
// 1 for 3*x mod pi = 0\ || | || 2 | // 0 for 6*x mod pi = 0\ || cos (3*x) | || | ||-1 + -------------- | || 2*cos(3*x) | || 2/ pi\ | ||---------------------------------- otherwise | || cos |3*x - --| | ||/ 2 \ | - |< \ 2 / | + |<| cos (3*x) | / pi\ | ||------------------- otherwise | |||1 + --------------|*cos|3*x - --| | || 2 | ||| 2/ pi\| \ 2 / | || cos (3*x) | ||| cos |3*x - --|| | || 1 + -------------- | ||\ \ 2 // | || 2/ pi\ | \\ / || cos |3*x - --| | \\ \ 2 / /
$$\left(\begin{cases} 0 & \text{for}\: 6 x \bmod \pi = 0 \\\frac{2 \cos{\left(3 x \right)}}{\left(\frac{\cos^{2}{\left(3 x \right)}}{\cos^{2}{\left(3 x - \frac{\pi}{2} \right)}} + 1\right) \cos{\left(3 x - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) - \left(\begin{cases} 1 & \text{for}\: 3 x \bmod \pi = 0 \\\frac{\frac{\cos^{2}{\left(3 x \right)}}{\cos^{2}{\left(3 x - \frac{\pi}{2} \right)}} - 1}{\frac{\cos^{2}{\left(3 x \right)}}{\cos^{2}{\left(3 x - \frac{\pi}{2} \right)}} + 1} & \text{otherwise} \end{cases}\right)$$
// 1 for 3*x mod pi = 0\
|| | // 0 for 6*x mod pi = 0\
|| 2/ pi\ | || |
|| sec |3*x - --| | || / pi\ |
|| \ 2 / | || 2*sec|3*x - --| |
||-1 + -------------- | || \ 2 / |
|| 2 | ||----------------------------- otherwise |
- |< sec (3*x) | + |
$$\left(\begin{cases} 0 & \text{for}\: 6 x \bmod \pi = 0 \\\frac{2 \sec{\left(3 x - \frac{\pi}{2} \right)}}{\left(1 + \frac{\sec^{2}{\left(3 x - \frac{\pi}{2} \right)}}{\sec^{2}{\left(3 x \right)}}\right) \sec{\left(3 x \right)}} & \text{otherwise} \end{cases}\right) - \left(\begin{cases} 1 & \text{for}\: 3 x \bmod \pi = 0 \\\frac{-1 + \frac{\sec^{2}{\left(3 x - \frac{\pi}{2} \right)}}{\sec^{2}{\left(3 x \right)}}}{1 + \frac{\sec^{2}{\left(3 x - \frac{\pi}{2} \right)}}{\sec^{2}{\left(3 x \right)}}} & \text{otherwise} \end{cases}\right)$$
// 1 for 3*x mod pi = 0\ || | || 2 | // 0 for 6*x mod pi = 0\ || csc (3*x) | || | ||-1 + -------------- | || 2*csc(3*x) | || 2/pi \ | ||---------------------------------- otherwise | || csc |-- - 3*x| | ||/ 2 \ | - |< \2 / | + |<| csc (3*x) | /pi \ | ||------------------- otherwise | |||1 + --------------|*csc|-- - 3*x| | || 2 | ||| 2/pi \| \2 / | || csc (3*x) | ||| csc |-- - 3*x|| | || 1 + -------------- | ||\ \2 // | || 2/pi \ | \\ / || csc |-- - 3*x| | \\ \2 / /
$$\left(\begin{cases} 0 & \text{for}\: 6 x \bmod \pi = 0 \\\frac{2 \csc{\left(3 x \right)}}{\left(\frac{\csc^{2}{\left(3 x \right)}}{\csc^{2}{\left(- 3 x + \frac{\pi}{2} \right)}} + 1\right) \csc{\left(- 3 x + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) - \left(\begin{cases} 1 & \text{for}\: 3 x \bmod \pi = 0 \\\frac{\frac{\csc^{2}{\left(3 x \right)}}{\csc^{2}{\left(- 3 x + \frac{\pi}{2} \right)}} - 1}{\frac{\csc^{2}{\left(3 x \right)}}{\csc^{2}{\left(- 3 x + \frac{\pi}{2} \right)}} + 1} & \text{otherwise} \end{cases}\right)$$
-Piecewise((1, Mod(3*x = pi, 0)), ((-1 + csc(3*x)^2/csc(pi/2 - 3*x)^2)/(1 + csc(3*x)^2/csc(pi/2 - 3*x)^2), True)) + Piecewise((0, Mod(6*x = pi, 0)), (2*csc(3*x)/((1 + csc(3*x)^2/csc(pi/2 - 3*x)^2)*csc(pi/2 - 3*x)), True))
Степени
[src]
-6*I*x 6*I*x / -6*I*x 6*I*x\
e e I*\- e + e /
- ------- - ------ - ----------------------
2 2 2
$$- \frac{e^{6 i x}}{2} - \frac{i \left(e^{6 i x} - e^{- 6 i x}\right)}{2} - \frac{e^{- 6 i x}}{2}$$
-exp(-6*i*x)/2 - exp(6*i*x)/2 - i*(-exp(-6*i*x) + exp(6*i*x))/2