Господин Экзамен
1/sin(2*x) если x=1
Решение
Подстановка условия
[src]
1/sin(2*x) при x = 1
подставляем
1 1*-------- sin(2*x)
$$1 \cdot \frac{1}{\sin{\left(2 x \right)}}$$
1 -------- sin(2*x)
$$\frac{1}{\sin{\left(2 x \right)}}$$
переменные
x = 1
$$x = 1$$
1 ---------- sin(2*(1))
$$\frac{1}{\sin{\left(2 (1) \right)}}$$
1 -------- sin(2*1)
$$\frac{1}{\sin{\left(2 \cdot 1 \right)}}$$
1 ------ sin(2)
$$\frac{1}{\sin{\left(2 \right)}}$$
1/sin(2)
Собрать выражение
[src]
csc(2*x)
$$\csc{\left(2 x \right)}$$
1 -------- sin(2*x)
$$\frac{1}{\sin{\left(2 x \right)}}$$
1/sin(2*x)
Степени
[src]
1 -------- sin(2*x)
$$\frac{1}{\sin{\left(2 x \right)}}$$
2*I ------------------ -2*I*x 2*I*x - e + e
$$\frac{2 i}{e^{2 i x} - e^{- 2 i x}}$$
2*i/(-exp(-2*i*x) + exp(2*i*x))
Тригонометрическая часть
[src]
csc(2*x)
$$\csc{\left(2 x \right)}$$
csc(pi - 2*x)
$$\csc{\left(- 2 x + \pi \right)}$$
1 -------- sin(2*x)
$$\frac{1}{\sin{\left(2 x \right)}}$$
/ pi\ sec|2*x - --| \ 2 /
$$\sec{\left(2 x - \frac{\pi}{2} \right)}$$
/pi \ sec|-- - 2*x| \2 /
$$\sec{\left(- 2 x + \frac{\pi}{2} \right)}$$
1 ------------- / pi\ cos|2*x - --| \ 2 /
$$\frac{1}{\cos{\left(2 x - \frac{\pi}{2} \right)}}$$
1 --------------- 2*cos(x)*sin(x)
$$\frac{1}{2 \sin{\left(x \right)} \cos{\left(x \right)}}$$
2 1 + tan (x) ----------- 2*tan(x)
$$\frac{\tan^{2}{\left(x \right)} + 1}{2 \tan{\left(x \right)}}$$
2 1 + cot (x) ----------- 2*cot(x)
$$\frac{\cot^{2}{\left(x \right)} + 1}{2 \cot{\left(x \right)}}$$
1 tan(x) -------- + ------ 2*tan(x) 2
$$\frac{\tan{\left(x \right)}}{2} + \frac{1}{2 \tan{\left(x \right)}}$$
/ 1 \
|1 + -------|*cot(x)
| 2 |
\ cot (x)/
--------------------
2
$$\frac{\left(1 + \frac{1}{\cot^{2}{\left(x \right)}}\right) \cot{\left(x \right)}}{2}$$
2/ pi\
1 + tan |x + --|
\ 4 /
-----------------
2/ pi\
-1 + tan |x + --|
\ 4 /
$$\frac{\tan^{2}{\left(x + \frac{\pi}{4} \right)} + 1}{\tan^{2}{\left(x + \frac{\pi}{4} \right)} - 1}$$
2/ pi\
1 + cot |x + --|
\ 4 /
----------------
2/ pi\
1 - cot |x + --|
\ 4 /
$$\frac{\cot^{2}{\left(x + \frac{\pi}{4} \right)} + 1}{- \cot^{2}{\left(x + \frac{\pi}{4} \right)} + 1}$$
/ zoo for 2*x mod pi = 0 < \csc(2*x) otherwise
$$\begin{cases} \tilde{\infty} & \text{for}\: 2 x \bmod \pi = 0 \\\csc{\left(2 x \right)} & \text{otherwise} \end{cases}$$
/ zoo for 2*x mod pi = 0 | < 1 |-------- otherwise \sin(2*x)
$$\begin{cases} \tilde{\infty} & \text{for}\: 2 x \bmod \pi = 0 \\\frac{1}{\sin{\left(2 x \right)}} & \text{otherwise} \end{cases}$$
/ zoo for 2*x mod pi = 0 | < / pi\ |sec|2*x - --| otherwise \ \ 2 /
$$\begin{cases} \tilde{\infty} & \text{for}\: 2 x \bmod \pi = 0 \\\sec{\left(2 x - \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}$$
/ 4 \
| 4*sin (x)|
|1 + ---------|*sin(2*x)
| 2 |
\ sin (2*x)/
------------------------
2
4*sin (x)
$$\frac{\left(\frac{4 \sin^{4}{\left(x \right)}}{\sin^{2}{\left(2 x \right)}} + 1\right) \sin{\left(2 x \right)}}{4 \sin^{2}{\left(x \right)}}$$
/ zoo for 2*x mod pi = 0 | | 1 <------------- otherwise | / pi\ |cos|2*x - --| \ \ 2 /
$$\begin{cases} \tilde{\infty} & \text{for}\: 2 x \bmod \pi = 0 \\\frac{1}{\cos{\left(2 x - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}$$
/ zoo for 2*x mod pi = 0 | | 2 <1 + cot (x) |----------- otherwise | 2*cot(x) \
$$\begin{cases} \tilde{\infty} & \text{for}\: 2 x \bmod \pi = 0 \\\frac{\cot^{2}{\left(x \right)} + 1}{2 \cot{\left(x \right)}} & \text{otherwise} \end{cases}$$
/ zoo for 2*x mod pi = 0 | | 2 <1 + tan (x) |----------- otherwise | 2*tan(x) \
$$\begin{cases} \tilde{\infty} & \text{for}\: 2 x \bmod \pi = 0 \\\frac{\tan^{2}{\left(x \right)} + 1}{2 \tan{\left(x \right)}} & \text{otherwise} \end{cases}$$
/ 2/ pi\\
| cos |x - --||
| \ 2 /|
|1 + ------------|*cos(x)
| 2 |
\ cos (x) /
-------------------------
/ pi\
2*cos|x - --|
\ 2 /
$$\frac{\left(1 + \frac{\cos^{2}{\left(x - \frac{\pi}{2} \right)}}{\cos^{2}{\left(x \right)}}\right) \cos{\left(x \right)}}{2 \cos{\left(x - \frac{\pi}{2} \right)}}$$
/ 2 \
| sec (x) | / pi\
|1 + ------------|*sec|x - --|
| 2/ pi\| \ 2 /
| sec |x - --||
\ \ 2 //
------------------------------
2*sec(x)
$$\frac{\left(\frac{\sec^{2}{\left(x \right)}}{\sec^{2}{\left(x - \frac{\pi}{2} \right)}} + 1\right) \sec{\left(x - \frac{\pi}{2} \right)}}{2 \sec{\left(x \right)}}$$
/ 2/pi \\
| csc |-- - x||
| \2 /|
|1 + ------------|*csc(x)
| 2 |
\ csc (x) /
-------------------------
/pi \
2*csc|-- - x|
\2 /
$$\frac{\left(1 + \frac{\csc^{2}{\left(- x + \frac{\pi}{2} \right)}}{\csc^{2}{\left(x \right)}}\right) \csc{\left(x \right)}}{2 \csc{\left(- x + \frac{\pi}{2} \right)}}$$
/ zoo for 2*x mod pi = 0 | |/ 1 \ ||1 + -------|*tan(x) <| 2 | |\ tan (x)/ |-------------------- otherwise | 2 \
$$\begin{cases} \tilde{\infty} & \text{for}\: 2 x \bmod \pi = 0 \\\frac{\left(1 + \frac{1}{\tan^{2}{\left(x \right)}}\right) \tan{\left(x \right)}}{2} & \text{otherwise} \end{cases}$$
/ / 3*pi\ | 1 for |2*x + ----| mod 2*pi = 0 | \ 2 / | < 1 |------ + tan(x) |tan(x) |--------------- otherwise \ 2
$$\begin{cases} 1 & \text{for}\: \left(2 x + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\frac{\tan{\left(x \right)} + \frac{1}{\tan{\left(x \right)}}}{2} & \text{otherwise} \end{cases}$$
/ / 3*pi\ | 1 for |2*x + ----| mod 2*pi = 0 | \ 2 / | | 2/ pi\ < 1 + tan |x + --| | \ 4 / |----------------- otherwise | 2/ pi\ |-1 + tan |x + --| \ \ 4 /
$$\begin{cases} 1 & \text{for}\: \left(2 x + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\frac{\tan^{2}{\left(x + \frac{\pi}{4} \right)} + 1}{\tan^{2}{\left(x + \frac{\pi}{4} \right)} - 1} & \text{otherwise} \end{cases}$$
/ zoo for 2*x mod pi = 0 | |/ zoo for 2*x mod pi = 0 <| |< 1 otherwise ||-------- otherwise \\sin(2*x)
$$\begin{cases} \tilde{\infty} & \text{for}\: 2 x \bmod \pi = 0 \\\begin{cases} \tilde{\infty} & \text{for}\: 2 x \bmod \pi = 0 \\\frac{1}{\sin{\left(2 x \right)}} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}$$
/ zoo for 2*x mod pi = 0 | | / 2 \ | 2 | sin (2*x)|
$$\begin{cases} \tilde{\infty} & \text{for}\: 2 x \bmod \pi = 0 \\\frac{\left(1 + \frac{\sin^{2}{\left(2 x \right)}}{4 \sin^{4}{\left(x \right)}}\right) \sin^{2}{\left(x \right)}}{\sin{\left(2 x \right)}} & \text{otherwise} \end{cases}$$
/ zoo for 2*x mod pi = 0 | |/ zoo for 2*x mod pi = 0 || <| 2 |<1 + cot (x) otherwise ||----------- otherwise || 2*cot(x) \\
$$\begin{cases} \tilde{\infty} & \text{for}\: 2 x \bmod \pi = 0 \\\begin{cases} \tilde{\infty} & \text{for}\: 2 x \bmod \pi = 0 \\\frac{\cot^{2}{\left(x \right)} + 1}{2 \cot{\left(x \right)}} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}$$
/ zoo for 2*x mod pi = 0 | |/ 2/ pi\\ || sec |x - --|| || \ 2 /| ||1 + ------------|*sec(x) <| 2 | |\ sec (x) / |------------------------- otherwise | / pi\ | 2*sec|x - --| | \ 2 / \
$$\begin{cases} \tilde{\infty} & \text{for}\: 2 x \bmod \pi = 0 \\\frac{\left(1 + \frac{\sec^{2}{\left(x - \frac{\pi}{2} \right)}}{\sec^{2}{\left(x \right)}}\right) \sec{\left(x \right)}}{2 \sec{\left(x - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}$$
/ zoo for 2*x mod pi = 0 | |/ 2 \ || cos (x) | / pi\ ||1 + ------------|*cos|x - --| <| 2/ pi\| \ 2 / || cos |x - --|| |\ \ 2 // |------------------------------ otherwise | 2*cos(x) \
$$\begin{cases} \tilde{\infty} & \text{for}\: 2 x \bmod \pi = 0 \\\frac{\left(\frac{\cos^{2}{\left(x \right)}}{\cos^{2}{\left(x - \frac{\pi}{2} \right)}} + 1\right) \cos{\left(x - \frac{\pi}{2} \right)}}{2 \cos{\left(x \right)}} & \text{otherwise} \end{cases}$$
/ zoo for 2*x mod pi = 0 | |/ 2 \ || csc (x) | /pi \ ||1 + ------------|*csc|-- - x| <| 2/pi \| \2 / || csc |-- - x|| |\ \2 // |------------------------------ otherwise | 2*csc(x) \
$$\begin{cases} \tilde{\infty} & \text{for}\: 2 x \bmod \pi = 0 \\\frac{\left(\frac{\csc^{2}{\left(x \right)}}{\csc^{2}{\left(- x + \frac{\pi}{2} \right)}} + 1\right) \csc{\left(- x + \frac{\pi}{2} \right)}}{2 \csc{\left(x \right)}} & \text{otherwise} \end{cases}$$
Piecewise((±oo, Mod(2*x = pi, 0)), ((1 + csc(x)^2/csc(pi/2 - x)^2)*csc(pi/2 - x)/(2*csc(x)), True))
Объединение рациональных выражений
[src]
1 -------- sin(2*x)
$$\frac{1}{\sin{\left(2 x \right)}}$$
1/sin(2*x)