Господин Экзамен
Общий знаменатель (log(sin(2*x))/3-log(4*sin(2*x)^2+3)/6)/2
Решение
Вы ввели
[src]
/ 2 \
log(sin(2*x)) log\4*sin (2*x) + 3/
------------- - --------------------
3 6
------------------------------------
2
$$\frac{- \frac{\log{\left(4 \sin^{2}{\left(2 x \right)} + 3 \right)}}{6} + \frac{\log{\left(\sin{\left(2 x \right)} \right)}}{3}}{2}$$
-log(4*sin(2*x)^2 + 3)/6 + log(sin(2*x))/3/2
Общее упрощение
[src]
/ 2 \
log\3 + 4*sin (2*x)/ log(sin(2*x))
- -------------------- + -------------
12 6
$$- \frac{\log{\left(4 \sin^{2}{\left(2 x \right)} + 3 \right)}}{12} + \frac{\log{\left(\sin{\left(2 x \right)} \right)}}{6}$$
-log(3 + 4*sin(2*x)^2)/12 + log(sin(2*x))/6
Собрать выражение
[src]
log(5 - 2*cos(4*x)) log(sin(2*x))
- ------------------- + -------------
6 6
$$- \frac{\log{\left(- 2 \cos{\left(4 x \right)} + 5 \right)}}{6} + \frac{\log{\left(\sin{\left(2 x \right)} \right)}}{6}$$
/ 2 \
log\3 + 4*sin (2*x)/ log(sin(2*x))
- -------------------- + -------------
6 6
$$- \frac{\log{\left(4 \sin^{2}{\left(2 x \right)} + 3 \right)}}{6} + \frac{\log{\left(\sin{\left(2 x \right)} \right)}}{6}$$
/ 2 \
log\4*sin (2*x) + 3/ log(sin(2*x))
- -------------------- + -------------
12 6
$$- \frac{\log{\left(4 \sin^{2}{\left(2 x \right)} + 3 \right)}}{12} + \frac{\log{\left(\sin{\left(2 x \right)} \right)}}{6}$$
-log(4*sin(2*x)^2 + 3)/12 + log(sin(2*x))/6
Тригонометрическая часть
[src]
log(5 - 2*cos(4*x)) log(sin(2*x))
- ------------------- + -------------
12 6
$$- \frac{\log{\left(- 2 \cos{\left(4 x \right)} + 5 \right)}}{12} + \frac{\log{\left(\sin{\left(2 x \right)} \right)}}{6}$$
/ 2 \
log\3 + 4*sin (2*x)/ log(sin(2*x))
- -------------------- + -------------
12 6
$$- \frac{\log{\left(4 \sin^{2}{\left(2 x \right)} + 3 \right)}}{12} + \frac{\log{\left(\sin{\left(2 x \right)} \right)}}{6}$$
/ 2 \
log|5 - --------|
\ sec(4*x)/ log(sin(2*x))
- ----------------- + -------------
12 6
$$- \frac{\log{\left(5 - \frac{2}{\sec{\left(4 x \right)}} \right)}}{12} + \frac{\log{\left(\sin{\left(2 x \right)} \right)}}{6}$$
/ 4 \
log|3 + ---------|
| 2 |
\ csc (2*x)/ log(sin(2*x))
- ------------------ + -------------
12 6
$$- \frac{\log{\left(3 + \frac{4}{\csc^{2}{\left(2 x \right)}} \right)}}{12} + \frac{\log{\left(\sin{\left(2 x \right)} \right)}}{6}$$
/ /pi \\
log|5 - 2*sin|-- + 4*x||
\ \2 // log(sin(2*x))
- ------------------------ + -------------
12 6
$$- \frac{\log{\left(- 2 \sin{\left(4 x + \frac{\pi}{2} \right)} + 5 \right)}}{12} + \frac{\log{\left(\sin{\left(2 x \right)} \right)}}{6}$$
/ / pi\\
log|cos|2*x - --||
log(5 - 2*cos(4*x)) \ \ 2 //
- ------------------- + ------------------
12 6
$$- \frac{\log{\left(- 2 \cos{\left(4 x \right)} + 5 \right)}}{12} + \frac{\log{\left(\cos{\left(2 x - \frac{\pi}{2} \right)} \right)}}{6}$$
/ 4 \
log|3 + ---------| / 1 \
| 2 | log|--------|
\ csc (2*x)/ \csc(2*x)/
- ------------------ + -------------
12 6
$$- \frac{\log{\left(3 + \frac{4}{\csc^{2}{\left(2 x \right)}} \right)}}{12} + \frac{\log{\left(\frac{1}{\csc{\left(2 x \right)}} \right)}}{6}$$
log(3 + 2*(1 - cos(4*x))) log(sin(2*x))
- ------------------------- + -------------
12 6
$$- \frac{\log{\left(2 \cdot \left(- \cos{\left(4 x \right)} + 1\right) + 3 \right)}}{12} + \frac{\log{\left(\sin{\left(2 x \right)} \right)}}{6}$$
/ 2 \
log|5 - -------------|
| /pi \|
| csc|-- - 4*x||
\ \2 // log(sin(2*x))
- ---------------------- + -------------
12 6
$$- \frac{\log{\left(5 - \frac{2}{\csc{\left(- 4 x + \frac{\pi}{2} \right)}} \right)}}{12} + \frac{\log{\left(\sin{\left(2 x \right)} \right)}}{6}$$
/ 1 \
log|-------------|
/ 2 \ | / pi\|
log|5 - --------| |sec|2*x - --||
\ sec(4*x)/ \ \ 2 //
- ----------------- + ------------------
12 6
$$- \frac{\log{\left(5 - \frac{2}{\sec{\left(4 x \right)}} \right)}}{12} + \frac{\log{\left(\frac{1}{\sec{\left(2 x - \frac{\pi}{2} \right)}} \right)}}{6}$$
/ 2 \
log|5 - -------------|
| /pi \| / 1 \
| csc|-- - 4*x|| log|--------|
\ \2 // \csc(2*x)/
- ---------------------- + -------------
12 6
$$- \frac{\log{\left(5 - \frac{2}{\csc{\left(- 4 x + \frac{\pi}{2} \right)}} \right)}}{12} + \frac{\log{\left(\frac{1}{\csc{\left(2 x \right)}} \right)}}{6}$$
/ 2/ pi\\ / / pi\\
log|3 + 4*cos |2*x - --|| log|cos|2*x - --||
\ \ 2 // \ \ 2 //
- ------------------------- + ------------------
12 6
$$- \frac{\log{\left(4 \cos^{2}{\left(2 x - \frac{\pi}{2} \right)} + 3 \right)}}{12} + \frac{\log{\left(\cos{\left(2 x - \frac{\pi}{2} \right)} \right)}}{6}$$
/ 4 2 \
log\3 - 16*cos (x) + 16*cos (x)/ log(sin(2*x))
- -------------------------------- + -------------
12 6
$$- \frac{\log{\left(- 16 \cos^{4}{\left(x \right)} + 16 \cos^{2}{\left(x \right)} + 3 \right)}}{12} + \frac{\log{\left(\sin{\left(2 x \right)} \right)}}{6}$$
/ 4 2 \
log\3 - 16*sin (x) + 16*sin (x)/ log(sin(2*x))
- -------------------------------- + -------------
12 6
$$- \frac{\log{\left(- 16 \sin^{4}{\left(x \right)} + 16 \sin^{2}{\left(x \right)} + 3 \right)}}{12} + \frac{\log{\left(\sin{\left(2 x \right)} \right)}}{6}$$
/ 4 \
log|3 + --------------| / 1 \
| 2 | log|-------------|
\ csc (pi - 2*x)/ \csc(pi - 2*x)/
- ----------------------- + ------------------
12 6
$$- \frac{\log{\left(3 + \frac{4}{\csc^{2}{\left(- 2 x + \pi \right)}} \right)}}{12} + \frac{\log{\left(\frac{1}{\csc{\left(- 2 x + \pi \right)}} \right)}}{6}$$
/ 4 \ / 1 \
log|3 + --------------| log|-------------|
| 2/ pi\| | / pi\|
| sec |2*x - --|| |sec|2*x - --||
\ \ 2 // \ \ 2 //
- ----------------------- + ------------------
12 6
$$- \frac{\log{\left(3 + \frac{4}{\sec^{2}{\left(2 x - \frac{\pi}{2} \right)}} \right)}}{12} + \frac{\log{\left(\frac{1}{\sec{\left(2 x - \frac{\pi}{2} \right)}} \right)}}{6}$$
/ 4 \ / 1 \
log|3 + --------------| log|-------------|
| 2/pi \| | / pi\|
| sec |-- - 2*x|| |sec|2*x - --||
\ \2 // \ \ 2 //
- ----------------------- + ------------------
12 6
$$- \frac{\log{\left(3 + \frac{4}{\sec^{2}{\left(- 2 x + \frac{\pi}{2} \right)}} \right)}}{12} + \frac{\log{\left(\frac{1}{\sec{\left(2 x - \frac{\pi}{2} \right)}} \right)}}{6}$$
/ 4 \ / 1 \
log|3 + --------------| log|-------------|
| 2/pi \| | /pi \|
| sec |-- - 2*x|| |sec|-- - 2*x||
\ \2 // \ \2 //
- ----------------------- + ------------------
12 6
$$- \frac{\log{\left(3 + \frac{4}{\sec^{2}{\left(- 2 x + \frac{\pi}{2} \right)}} \right)}}{12} + \frac{\log{\left(\frac{1}{\sec{\left(- 2 x + \frac{\pi}{2} \right)}} \right)}}{6}$$
/ 2 \
| 16*tan (x) |
log|3 + --------------| / 2*tan(x) \
| 2| log|-----------|
| / 2 \ | | 2 |
\ \1 + tan (x)/ / \1 + tan (x)/
- ----------------------- + ----------------
12 6
$$\frac{\log{\left(\frac{2 \tan{\left(x \right)}}{\tan^{2}{\left(x \right)} + 1} \right)}}{6} - \frac{\log{\left(3 + \frac{16 \tan^{2}{\left(x \right)}}{\left(\tan^{2}{\left(x \right)} + 1\right)^{2}} \right)}}{12}$$
/ 2 \
| 16*cot (x) |
log|3 + --------------| / 2*cot(x) \
| 2| log|-----------|
| / 2 \ | | 2 |
\ \1 + cot (x)/ / \1 + cot (x)/
- ----------------------- + ----------------
12 6
$$\frac{\log{\left(\frac{2 \cot{\left(x \right)}}{\cot^{2}{\left(x \right)} + 1} \right)}}{6} - \frac{\log{\left(3 + \frac{16 \cot^{2}{\left(x \right)}}{\left(\cot^{2}{\left(x \right)} + 1\right)^{2}} \right)}}{12}$$
/ / 2 \\
| 2*\1 - tan (2*x)/| / 2*tan(x) \
log|5 - -----------------| log|-----------|
| 2 | | 2 |
\ 1 + tan (2*x) / \1 + tan (x)/
- -------------------------- + ----------------
12 6
$$\frac{\log{\left(\frac{2 \tan{\left(x \right)}}{\tan^{2}{\left(x \right)} + 1} \right)}}{6} - \frac{\log{\left(- \frac{2 \cdot \left(- \tan^{2}{\left(2 x \right)} + 1\right)}{\tan^{2}{\left(2 x \right)} + 1} + 5 \right)}}{12}$$
// 0 for 2*x mod pi = 0\
log|< |
log(3 + 2*(1 - cos(4*x))) \\sin(2*x) otherwise /
- ------------------------- + ----------------------------------
12 6
$$- \frac{\log{\left(2 \cdot \left(- \cos{\left(4 x \right)} + 1\right) + 3 \right)}}{12} + \left(\frac{\log{\left(\begin{cases} 0 & \text{for}\: 2 x \bmod \pi = 0 \\\sin{\left(2 x \right)} & \text{otherwise} \end{cases} \right)}}{6}\right)$$
/ 2 \
| / 2 2 \ 2 2 |
log\5 - 2*\cos (x) - sin (x)/ + 8*cos (x)*sin (x)/ log(2*cos(x)*sin(x))
- --------------------------------------------------- + --------------------
12 6
$$\frac{\log{\left(2 \sin{\left(x \right)} \cos{\left(x \right)} \right)}}{6} - \frac{\log{\left(8 \sin^{2}{\left(x \right)} \cos^{2}{\left(x \right)} - 2 \left(- \sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right)^{2} + 5 \right)}}{12}$$
/ 16 \
log|3 + ----------------------| / 2 \
| 2 | log|--------------------|
| / 1 \ 2 | |/ 1 \ |
| |1 + -------| *cot (x)| ||1 + -------|*cot(x)|
| | 2 | | || 2 | |
\ \ cot (x)/ / \\ cot (x)/ /
- ------------------------------- + -------------------------
12 6
$$\frac{\log{\left(\frac{2}{\left(1 + \frac{1}{\cot^{2}{\left(x \right)}}\right) \cot{\left(x \right)}} \right)}}{6} - \frac{\log{\left(3 + \frac{16}{\left(1 + \frac{1}{\cot^{2}{\left(x \right)}}\right)^{2} \cot^{2}{\left(x \right)}} \right)}}{12}$$
/ 2\
| / 2/ pi\\ |
| 4*|-1 + tan |x + --|| | / 2/ pi\\
| \ \ 4 // | |-1 + tan |x + --||
log|3 + ----------------------| | \ 4 /|
| 2 | log|-----------------|
| / 2/ pi\\ | | 2/ pi\|
| |1 + tan |x + --|| | | 1 + tan |x + --||
\ \ \ 4 // / \ \ 4 //
- ------------------------------- + ----------------------
12 6
$$\frac{\log{\left(\frac{\tan^{2}{\left(x + \frac{\pi}{4} \right)} - 1}{\tan^{2}{\left(x + \frac{\pi}{4} \right)} + 1} \right)}}{6} - \frac{\log{\left(\frac{4 \left(\tan^{2}{\left(x + \frac{\pi}{4} \right)} - 1\right)^{2}}{\left(\tan^{2}{\left(x + \frac{\pi}{4} \right)} + 1\right)^{2}} + 3 \right)}}{12}$$
/ 2\
| / 2/ pi\\ |
| 4*|1 - cot |x + --|| | / 2/ pi\\
| \ \ 4 // | |1 - cot |x + --||
log|3 + ---------------------| | \ 4 /|
| 2 | log|----------------|
| / 2/ pi\\ | | 2/ pi\|
| |1 + cot |x + --|| | |1 + cot |x + --||
\ \ \ 4 // / \ \ 4 //
- ------------------------------ + ---------------------
12 6
$$\frac{\log{\left(\frac{- \cot^{2}{\left(x + \frac{\pi}{4} \right)} + 1}{\cot^{2}{\left(x + \frac{\pi}{4} \right)} + 1} \right)}}{6} - \frac{\log{\left(\frac{4 \left(- \cot^{2}{\left(x + \frac{\pi}{4} \right)} + 1\right)^{2}}{\left(\cot^{2}{\left(x + \frac{\pi}{4} \right)} + 1\right)^{2}} + 3 \right)}}{12}$$
// 0 for 2*x mod pi = 0\
/ 2 \ || |
| 16*cot (x) | || 2*cot(x) |
log|3 + --------------| log|<----------- otherwise |
| 2| || 2 |
| / 2 \ | ||1 + cot (x) |
\ \1 + cot (x)/ / \\ /
- ----------------------- + -------------------------------------
12 6
$$- \frac{\log{\left(3 + \frac{16 \cot^{2}{\left(x \right)}}{\left(\cot^{2}{\left(x \right)} + 1\right)^{2}} \right)}}{12} + \left(\frac{\log{\left(\begin{cases} 0 & \text{for}\: 2 x \bmod \pi = 0 \\\frac{2 \cot{\left(x \right)}}{\cot^{2}{\left(x \right)} + 1} & \text{otherwise} \end{cases} \right)}}{6}\right)$$
/ // 0 for 2*x mod pi = 0\\
| || ||
log|3 + 4*|< 2 || // 0 for 2*x mod pi = 0\
| ||sin (2*x) otherwise || log|< |
\ \\ // \\sin(2*x) otherwise /
- ------------------------------------------- + ----------------------------------
12 6
$$\left(- \frac{\log{\left(\left(4 \left(\begin{cases} 0 & \text{for}\: 2 x \bmod \pi = 0 \\\sin^{2}{\left(2 x \right)} & \text{otherwise} \end{cases}\right)\right) + 3 \right)}}{12}\right) + \left(\frac{\log{\left(\begin{cases} 0 & \text{for}\: 2 x \bmod \pi = 0 \\\sin{\left(2 x \right)} & \text{otherwise} \end{cases} \right)}}{6}\right)$$
/ // 0 for 2*x mod pi = 0\\
| || ||
log|3 + 4*|<1 - cos(4*x) || // 0 for 2*x mod pi = 0\
| ||------------ otherwise || log|< |
\ \\ 2 // \\sin(2*x) otherwise /
- ---------------------------------------------- + ----------------------------------
12 6
$$\left(- \frac{\log{\left(\left(4 \left(\begin{cases} 0 & \text{for}\: 2 x \bmod \pi = 0 \\\frac{- \cos{\left(4 x \right)} + 1}{2} & \text{otherwise} \end{cases}\right)\right) + 3 \right)}}{12}\right) + \left(\frac{\log{\left(\begin{cases} 0 & \text{for}\: 2 x \bmod \pi = 0 \\\sin{\left(2 x \right)} & \text{otherwise} \end{cases} \right)}}{6}\right)$$
/ 4 \
| 64*sin (x) | / 2 \
log|3 + --------------------------| | 4*sin (x) |
| 2 | log|------------------------|
| / 4 \ | |/ 4 \ |
| | 4*sin (x)| 2 | || 4*sin (x)| |
| |1 + ---------| *sin (2*x)| ||1 + ---------|*sin(2*x)|
| | 2 | | || 2 | |
\ \ sin (2*x)/ / \\ sin (2*x)/ /
- ----------------------------------- + -----------------------------
12 6
$$\frac{\log{\left(\frac{4 \sin^{2}{\left(x \right)}}{\left(\frac{4 \sin^{4}{\left(x \right)}}{\sin^{2}{\left(2 x \right)}} + 1\right) \sin{\left(2 x \right)}} \right)}}{6} - \frac{\log{\left(3 + \frac{64 \sin^{4}{\left(x \right)}}{\left(\frac{4 \sin^{4}{\left(x \right)}}{\sin^{2}{\left(2 x \right)}} + 1\right)^{2} \sin^{2}{\left(2 x \right)}} \right)}}{12}$$
/ // 0 for 2*x mod pi = 0\\ // 0 for 2*x mod pi = 0\
| || || || |
log|3 + 4*|<1 cos(4*x) || log|< / pi\ |
| ||- - -------- otherwise || ||cos|2*x - --| otherwise |
\ \\2 2 // \\ \ 2 / /
- ---------------------------------------------- + ---------------------------------------
12 6
$$\left(- \frac{\log{\left(\left(4 \left(\begin{cases} 0 & \text{for}\: 2 x \bmod \pi = 0 \\- \frac{\cos{\left(4 x \right)}}{2} + \frac{1}{2} & \text{otherwise} \end{cases}\right)\right) + 3 \right)}}{12}\right) + \left(\frac{\log{\left(\begin{cases} 0 & \text{for}\: 2 x \bmod \pi = 0 \\\cos{\left(2 x - \frac{\pi}{2} \right)} & \text{otherwise} \end{cases} \right)}}{6}\right)$$
/ // 0 for 2*x mod pi = 0\\
| || ||
| || /pi \ ||
log|3 + 4*|< sin|-- + 4*x| ||
| ||1 \2 / || // 0 for 2*x mod pi = 0\
| ||- - ------------- otherwise || log|< |
\ \\2 2 // \\sin(2*x) otherwise /
- --------------------------------------------------- + ----------------------------------
12 6
$$\left(- \frac{\log{\left(\left(4 \left(\begin{cases} 0 & \text{for}\: 2 x \bmod \pi = 0 \\- \frac{\sin{\left(4 x + \frac{\pi}{2} \right)}}{2} + \frac{1}{2} & \text{otherwise} \end{cases}\right)\right) + 3 \right)}}{12}\right) + \left(\frac{\log{\left(\begin{cases} 0 & \text{for}\: 2 x \bmod \pi = 0 \\\sin{\left(2 x \right)} & \text{otherwise} \end{cases} \right)}}{6}\right)$$
/ // 1 for 2*x mod pi = 0\\
| || ||
log|5 - 2*|< 2 / 2 \ || // 0 for 2*x mod pi = 0\
| ||sin (2*x)*\-1 + cot (2*x)/ otherwise || log|< |
\ \\ // \\sin(2*x) otherwise /
- ------------------------------------------------------------ + ----------------------------------
12 6
$$\left(- \frac{\log{\left(\left(- 2 \left(\begin{cases} 1 & \text{for}\: 2 x \bmod \pi = 0 \\\left(\cot^{2}{\left(2 x \right)} - 1\right) \sin^{2}{\left(2 x \right)} & \text{otherwise} \end{cases}\right)\right) + 5 \right)}}{12}\right) + \left(\frac{\log{\left(\begin{cases} 0 & \text{for}\: 2 x \bmod \pi = 0 \\\sin{\left(2 x \right)} & \text{otherwise} \end{cases} \right)}}{6}\right)$$
// 0 for 2*x mod pi = 0\
|| |
/ // 0 for 2*x mod pi = 0\\ || 1 |
| || || log|<------------- otherwise |
log|3 + 4*|<1 1 || || / pi\ |
| ||- - ---------- otherwise || ||sec|2*x - --| |
\ \\2 2*sec(4*x) // \\ \ 2 / /
- ------------------------------------------------ + ---------------------------------------
12 6
$$\left(- \frac{\log{\left(\left(4 \left(\begin{cases} 0 & \text{for}\: 2 x \bmod \pi = 0 \\\frac{1}{2} - \frac{1}{2 \sec{\left(4 x \right)}} & \text{otherwise} \end{cases}\right)\right) + 3 \right)}}{12}\right) + \left(\frac{\log{\left(\begin{cases} 0 & \text{for}\: 2 x \bmod \pi = 0 \\\frac{1}{\sec{\left(2 x - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases} \right)}}{6}\right)$$
/ // 0 for 2*x mod pi = 0\\
| || ||
| ||1 1 || // 0 for 2*x mod pi = 0\
log|3 + 4*|<- - --------------- otherwise || || |
| ||2 /pi \ || log|< 1 |
| || 2*csc|-- - 4*x| || ||-------- otherwise |
\ \\ \2 / // \\csc(2*x) /
- ----------------------------------------------------- + ----------------------------------
12 6
$$\left(- \frac{\log{\left(\left(4 \left(\begin{cases} 0 & \text{for}\: 2 x \bmod \pi = 0 \\\frac{1}{2} - \frac{1}{2 \csc{\left(- 4 x + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) + 3 \right)}}{12}\right) + \left(\frac{\log{\left(\begin{cases} 0 & \text{for}\: 2 x \bmod \pi = 0 \\\frac{1}{\csc{\left(2 x \right)}} & \text{otherwise} \end{cases} \right)}}{6}\right)$$
/ 2 \
| 16*sec (x) |
log|3 + --------------------------------| / 2*sec(x) \
| 2 | log|------------------------------|
| / 2 \ | |/ 2 \ |
| | sec (x) | 2/ pi\| || sec (x) | / pi\|
| |1 + ------------| *sec |x - --|| ||1 + ------------|*sec|x - --||
| | 2/ pi\| \ 2 /| || 2/ pi\| \ 2 /|
| | sec |x - --|| | || sec |x - --|| |
\ \ \ 2 // / \\ \ 2 // /
- ----------------------------------------- + -----------------------------------
12 6
$$\frac{\log{\left(\frac{2 \sec{\left(x \right)}}{\left(\frac{\sec^{2}{\left(x \right)}}{\sec^{2}{\left(x - \frac{\pi}{2} \right)}} + 1\right) \sec{\left(x - \frac{\pi}{2} \right)}} \right)}}{6} - \frac{\log{\left(3 + \frac{16 \sec^{2}{\left(x \right)}}{\left(\frac{\sec^{2}{\left(x \right)}}{\sec^{2}{\left(x - \frac{\pi}{2} \right)}} + 1\right)^{2} \sec^{2}{\left(x - \frac{\pi}{2} \right)}} \right)}}{12}$$
/ 2/ pi\ \
| 16*cos |x - --| | / / pi\ \
| \ 2 / | | 2*cos|x - --| |
log|3 + ---------------------------| | \ 2 / |
| 2 | log|-------------------------|
| / 2/ pi\\ | |/ 2/ pi\\ |
| | cos |x - --|| | || cos |x - --|| |
| | \ 2 /| 2 | || \ 2 /| |
| |1 + ------------| *cos (x)| ||1 + ------------|*cos(x)|
| | 2 | | || 2 | |
\ \ cos (x) / / \\ cos (x) / /
- ------------------------------------ + ------------------------------
12 6
$$\frac{\log{\left(\frac{2 \cos{\left(x - \frac{\pi}{2} \right)}}{\left(1 + \frac{\cos^{2}{\left(x - \frac{\pi}{2} \right)}}{\cos^{2}{\left(x \right)}}\right) \cos{\left(x \right)}} \right)}}{6} - \frac{\log{\left(3 + \frac{16 \cos^{2}{\left(x - \frac{\pi}{2} \right)}}{\left(1 + \frac{\cos^{2}{\left(x - \frac{\pi}{2} \right)}}{\cos^{2}{\left(x \right)}}\right)^{2} \cos^{2}{\left(x \right)}} \right)}}{12}$$
/ 2/pi \ \
| 16*csc |-- - x| | / /pi \ \
| \2 / | | 2*csc|-- - x| |
log|3 + ---------------------------| | \2 / |
| 2 | log|-------------------------|
| / 2/pi \\ | |/ 2/pi \\ |
| | csc |-- - x|| | || csc |-- - x|| |
| | \2 /| 2 | || \2 /| |
| |1 + ------------| *csc (x)| ||1 + ------------|*csc(x)|
| | 2 | | || 2 | |
\ \ csc (x) / / \\ csc (x) / /
- ------------------------------------ + ------------------------------
12 6
$$\frac{\log{\left(\frac{2 \csc{\left(- x + \frac{\pi}{2} \right)}}{\left(1 + \frac{\csc^{2}{\left(- x + \frac{\pi}{2} \right)}}{\csc^{2}{\left(x \right)}}\right) \csc{\left(x \right)}} \right)}}{6} - \frac{\log{\left(3 + \frac{16 \csc^{2}{\left(- x + \frac{\pi}{2} \right)}}{\left(1 + \frac{\csc^{2}{\left(- x + \frac{\pi}{2} \right)}}{\csc^{2}{\left(x \right)}}\right)^{2} \csc^{2}{\left(x \right)}} \right)}}{12}$$
/ // 1 for 2*x mod pi = 0\\ // 0 for 2*x mod pi = 0\
| || || || |
| || 2 || || 2*cot(x) |
log|5 - 2*|<-1 + cot (2*x) || log|<----------- otherwise |
| ||-------------- otherwise || || 2 |
| || 2 || ||1 + cot (x) |
\ \\1 + cot (2*x) // \\ /
- ------------------------------------------------ + -------------------------------------
12 6
$$\left(- \frac{\log{\left(\left(- 2 \left(\begin{cases} 1 & \text{for}\: 2 x \bmod \pi = 0 \\\frac{\cot^{2}{\left(2 x \right)} - 1}{\cot^{2}{\left(2 x \right)} + 1} & \text{otherwise} \end{cases}\right)\right) + 5 \right)}}{12}\right) + \left(\frac{\log{\left(\begin{cases} 0 & \text{for}\: 2 x \bmod \pi = 0 \\\frac{2 \cot{\left(x \right)}}{\cot^{2}{\left(x \right)} + 1} & \text{otherwise} \end{cases} \right)}}{6}\right)$$
/ // 0 for 2*x mod pi = 0\\
| || ||
| || 2 || // 0 for 2*x mod pi = 0\
| || 4*cot (x) || || |
log|3 + 4*|<-------------- otherwise || || 2*cot(x) |
| || 2 || log|<----------- otherwise |
| ||/ 2 \ || || 2 |
| ||\1 + cot (x)/ || ||1 + cot (x) |
\ \\ // \\ /
- ------------------------------------------------ + -------------------------------------
12 6
$$\left(- \frac{\log{\left(\left(4 \left(\begin{cases} 0 & \text{for}\: 2 x \bmod \pi = 0 \\\frac{4 \cot^{2}{\left(x \right)}}{\left(\cot^{2}{\left(x \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right)\right) + 3 \right)}}{12}\right) + \left(\frac{\log{\left(\begin{cases} 0 & \text{for}\: 2 x \bmod \pi = 0 \\\frac{2 \cot{\left(x \right)}}{\cot^{2}{\left(x \right)} + 1} & \text{otherwise} \end{cases} \right)}}{6}\right)$$
/ // / 3*pi\ \\
| || 1 for |2*x + ----| mod 2*pi = 0||
| || \ 2 / || // / 3*pi\ \
log|3 + 4*|< || || 1 for |2*x + ----| mod 2*pi = 0|
| || 4 2 || log|< \ 2 / |
| ||- 4*cos (x) + 4*cos (x) otherwise || || |
\ \\ // \\sin(2*x) otherwise /
- -------------------------------------------------------------------- + ---------------------------------------------
12 6
$$\left(- \frac{\log{\left(\left(4 \left(\begin{cases} 1 & \text{for}\: \left(2 x + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\- 4 \cos^{4}{\left(x \right)} + 4 \cos^{2}{\left(x \right)} & \text{otherwise} \end{cases}\right)\right) + 3 \right)}}{12}\right) + \left(\frac{\log{\left(\begin{cases} 1 & \text{for}\: \left(2 x + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\sin{\left(2 x \right)} & \text{otherwise} \end{cases} \right)}}{6}\right)$$
/ // 0 for 2*x mod pi = 0\\ // 0 for 2*x mod pi = 0\
| || || || |
| || 2 || || 2*tan(x) |
log|3 + 4*|<1 1 - tan (2*x) || log|<----------- otherwise |
| ||- - ----------------- otherwise || || 2 |
| ||2 / 2 \ || ||1 + tan (x) |
\ \\ 2*\1 + tan (2*x)/ // \\ /
- ------------------------------------------------------- + -------------------------------------
12 6
$$\left(- \frac{\log{\left(\left(4 \left(\begin{cases} 0 & \text{for}\: 2 x \bmod \pi = 0 \\- \frac{- \tan^{2}{\left(2 x \right)} + 1}{2 \left(\tan^{2}{\left(2 x \right)} + 1\right)} + \frac{1}{2} & \text{otherwise} \end{cases}\right)\right) + 3 \right)}}{12}\right) + \left(\frac{\log{\left(\begin{cases} 0 & \text{for}\: 2 x \bmod \pi = 0 \\\frac{2 \tan{\left(x \right)}}{\tan^{2}{\left(x \right)} + 1} & \text{otherwise} \end{cases} \right)}}{6}\right)$$
/ // 0 for 2*x mod pi = 0\\ // 0 for 2*x mod pi = 0\
| || || || |
| || 4 || || 2 |
| ||---------------------- otherwise || ||-------------------- otherwise |
log|3 + 4*|< 2 || log|
$$\left(- \frac{\log{\left(\left(4 \left(\begin{cases} 0 & \text{for}\: 2 x \bmod \pi = 0 \\\frac{4}{\left(1 + \frac{1}{\tan^{2}{\left(x \right)}}\right)^{2} \tan^{2}{\left(x \right)}} & \text{otherwise} \end{cases}\right)\right) + 3 \right)}}{12}\right) + \left(\frac{\log{\left(\begin{cases} 0 & \text{for}\: 2 x \bmod \pi = 0 \\\frac{2}{\left(1 + \frac{1}{\tan^{2}{\left(x \right)}}\right) \tan{\left(x \right)}} & \text{otherwise} \end{cases} \right)}}{6}\right)$$
/ // 0 for 2*x mod pi = 0\\
| || ||
| || 2 || // 0 for 2*x mod pi = 0\
| || sin (2*x) || || |
| ||------------------------ otherwise || || sin(2*x) |
log|3 + 4*|< 2 || ||----------------------- otherwise |
| ||/ 2 \ || log|
$$\left(- \frac{\log{\left(\left(4 \left(\begin{cases} 0 & \text{for}\: 2 x \bmod \pi = 0 \\\frac{\sin^{2}{\left(2 x \right)}}{\left(1 + \frac{\sin^{2}{\left(2 x \right)}}{4 \sin^{4}{\left(x \right)}}\right)^{2} \sin^{4}{\left(x \right)}} & \text{otherwise} \end{cases}\right)\right) + 3 \right)}}{12}\right) + \left(\frac{\log{\left(\begin{cases} 0 & \text{for}\: 2 x \bmod \pi = 0 \\\frac{\sin{\left(2 x \right)}}{\left(1 + \frac{\sin^{2}{\left(2 x \right)}}{4 \sin^{4}{\left(x \right)}}\right) \sin^{2}{\left(x \right)}} & \text{otherwise} \end{cases} \right)}}{6}\right)$$
/ // 0 for 2*x mod pi = 0\\
| || ||
| || / 1 for 2*x mod pi = 0 ||
| || | ||
log|3 + 4*|< < 2 / 2 \ || // 0 for 2*x mod pi = 0\
| || |sin (2*x)*\-1 + cot (2*x)/ otherwise || || |
| ||1 \ || log|
$$\left(- \frac{\log{\left(\left(4 \left(\begin{cases} 0 & \text{for}\: 2 x \bmod \pi = 0 \\\left(- \frac{\begin{cases} 1 & \text{for}\: 2 x \bmod \pi = 0 \\\left(\cot^{2}{\left(2 x \right)} - 1\right) \sin^{2}{\left(2 x \right)} & \text{otherwise} \end{cases}}{2}\right) + \frac{1}{2} & \text{otherwise} \end{cases}\right)\right) + 3 \right)}}{12}\right) + \left(\frac{\log{\left(\begin{cases} 0 & \text{for}\: 2 x \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: 2 x \bmod \pi = 0 \\\sin{\left(2 x \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases} \right)}}{6}\right)$$
/ // / 3*pi\ \\
| || 1 for |2*x + ----| mod 2*pi = 0||
| || \ 2 / || // / 3*pi\ \
| || || || 1 for |2*x + ----| mod 2*pi = 0|
| || 2 || || \ 2 / |
| ||/ 2/ pi\\ || || |
log|3 + 4*|<|-1 + tan |x + --|| || || 2/ pi\ |
| ||\ \ 4 // || log|<-1 + tan |x + --| |
| ||-------------------- otherwise || || \ 4 / |
| || 2 || ||----------------- otherwise |
| ||/ 2/ pi\\ || || 2/ pi\ |
| |||1 + tan |x + --|| || || 1 + tan |x + --| |
\ \\\ \ 4 // // \\ \ 4 / /
- ----------------------------------------------------------------- + ------------------------------------------------------
12 6
$$\left(- \frac{\log{\left(\left(4 \left(\begin{cases} 1 & \text{for}\: \left(2 x + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\frac{\left(\tan^{2}{\left(x + \frac{\pi}{4} \right)} - 1\right)^{2}}{\left(\tan^{2}{\left(x + \frac{\pi}{4} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right)\right) + 3 \right)}}{12}\right) + \left(\frac{\log{\left(\begin{cases} 1 & \text{for}\: \left(2 x + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\frac{\tan^{2}{\left(x + \frac{\pi}{4} \right)} - 1}{\tan^{2}{\left(x + \frac{\pi}{4} \right)} + 1} & \text{otherwise} \end{cases} \right)}}{6}\right)$$
/ // 0 for 2*x mod pi = 0\\
| || ||
| || / 1 for 2*x mod pi = 0 || // 0 for 2*x mod pi = 0\
| || | || || |
| || | 2 || ||/ 0 for 2*x mod pi = 0 |
log|3 + 4*|< <-1 + cot (2*x) || ||| |
| || |-------------- otherwise || log|<| 2*cot(x) |
| || | 2 || ||<----------- otherwise otherwise |
| ||1 \1 + cot (2*x) || ||| 2 |
| ||- - ----------------------------------- otherwise || |||1 + cot (x) |
\ \\2 2 // \\\ /
- ------------------------------------------------------------------------- + ----------------------------------------------------------
12 6
$$\left(- \frac{\log{\left(\left(4 \left(\begin{cases} 0 & \text{for}\: 2 x \bmod \pi = 0 \\\left(- \frac{\begin{cases} 1 & \text{for}\: 2 x \bmod \pi = 0 \\\frac{\cot^{2}{\left(2 x \right)} - 1}{\cot^{2}{\left(2 x \right)} + 1} & \text{otherwise} \end{cases}}{2}\right) + \frac{1}{2} & \text{otherwise} \end{cases}\right)\right) + 3 \right)}}{12}\right) + \left(\frac{\log{\left(\begin{cases} 0 & \text{for}\: 2 x \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: 2 x \bmod \pi = 0 \\\frac{2 \cot{\left(x \right)}}{\cot^{2}{\left(x \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases} \right)}}{6}\right)$$
/ // 0 for 2*x mod pi = 0\\ // 0 for 2*x mod pi = 0\
| || || || |
| || 2/ pi\ || || / pi\ |
| || 4*sec |x - --| || || 2*sec|x - --| |
| || \ 2 / || || \ 2 / |
| ||--------------------------- otherwise || ||------------------------- otherwise |
log|3 + 4*|< 2 || log|
$$\left(- \frac{\log{\left(\left(4 \left(\begin{cases} 0 & \text{for}\: 2 x \bmod \pi = 0 \\\frac{4 \sec^{2}{\left(x - \frac{\pi}{2} \right)}}{\left(1 + \frac{\sec^{2}{\left(x - \frac{\pi}{2} \right)}}{\sec^{2}{\left(x \right)}}\right)^{2} \sec^{2}{\left(x \right)}} & \text{otherwise} \end{cases}\right)\right) + 3 \right)}}{12}\right) + \left(\frac{\log{\left(\begin{cases} 0 & \text{for}\: 2 x \bmod \pi = 0 \\\frac{2 \sec{\left(x - \frac{\pi}{2} \right)}}{\left(1 + \frac{\sec^{2}{\left(x - \frac{\pi}{2} \right)}}{\sec^{2}{\left(x \right)}}\right) \sec{\left(x \right)}} & \text{otherwise} \end{cases} \right)}}{6}\right)$$
/ // 0 for 2*x mod pi = 0\\
| || ||
| || 2 || // 0 for 2*x mod pi = 0\
| || 4*cos (x) || || |
| ||-------------------------------- otherwise || || 2*cos(x) |
| || 2 || ||------------------------------ otherwise |
log|3 + 4*|
$$\left(- \frac{\log{\left(\left(4 \left(\begin{cases} 0 & \text{for}\: 2 x \bmod \pi = 0 \\\frac{4 \cos^{2}{\left(x \right)}}{\left(\frac{\cos^{2}{\left(x \right)}}{\cos^{2}{\left(x - \frac{\pi}{2} \right)}} + 1\right)^{2} \cos^{2}{\left(x - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) + 3 \right)}}{12}\right) + \left(\frac{\log{\left(\begin{cases} 0 & \text{for}\: 2 x \bmod \pi = 0 \\\frac{2 \cos{\left(x \right)}}{\left(\frac{\cos^{2}{\left(x \right)}}{\cos^{2}{\left(x - \frac{\pi}{2} \right)}} + 1\right) \cos{\left(x - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases} \right)}}{6}\right)$$
/ // 0 for 2*x mod pi = 0\\
| || ||
| || 2 || // 0 for 2*x mod pi = 0\
| || 4*csc (x) || || |
| ||-------------------------------- otherwise || || 2*csc(x) |
| || 2 || ||------------------------------ otherwise |
log|3 + 4*|
$$\left(- \frac{\log{\left(\left(4 \left(\begin{cases} 0 & \text{for}\: 2 x \bmod \pi = 0 \\\frac{4 \csc^{2}{\left(x \right)}}{\left(\frac{\csc^{2}{\left(x \right)}}{\csc^{2}{\left(- x + \frac{\pi}{2} \right)}} + 1\right)^{2} \csc^{2}{\left(- x + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) + 3 \right)}}{12}\right) + \left(\frac{\log{\left(\begin{cases} 0 & \text{for}\: 2 x \bmod \pi = 0 \\\frac{2 \csc{\left(x \right)}}{\left(\frac{\csc^{2}{\left(x \right)}}{\csc^{2}{\left(- x + \frac{\pi}{2} \right)}} + 1\right) \csc{\left(- x + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases} \right)}}{6}\right)$$
-log(3 + 4*Piecewise((0, Mod(2*x = pi, 0)), (4*csc(x)^2/((1 + csc(x)^2/csc(pi/2 - x)^2)^2*csc(pi/2 - x)^2), True)))/12 + log(Piecewise((0, Mod(2*x = pi, 0)), (2*csc(x)/((1 + csc(x)^2/csc(pi/2 - x)^2)*csc(pi/2 - x)), True)))/6
Объединение рациональных выражений
[src]
/ 2 \
- log\3 + 4*sin (2*x)/ + 2*log(sin(2*x))
----------------------------------------
12
$$\frac{- \log{\left(4 \sin^{2}{\left(2 x \right)} + 3 \right)} + 2 \log{\left(\sin{\left(2 x \right)} \right)}}{12}$$
(-log(3 + 4*sin(2*x)^2) + 2*log(sin(2*x)))/12
Рациональный знаменатель
[src]
/ 2 \
- log\3 + 4*sin (2*x)/ + 2*log(sin(2*x))
----------------------------------------
12
$$\frac{- \log{\left(4 \sin^{2}{\left(2 x \right)} + 3 \right)} + 2 \log{\left(\sin{\left(2 x \right)} \right)}}{12}$$
/ 2 \
log\3 + 4*sin (2*x)/ log(sin(2*x))
- -------------------- + -------------
12 6
$$- \frac{\log{\left(4 \sin^{2}{\left(2 x \right)} + 3 \right)}}{12} + \frac{\log{\left(\sin{\left(2 x \right)} \right)}}{6}$$
-log(3 + 4*sin(2*x)^2)/12 + log(sin(2*x))/6
Численный ответ
[src]
0.166666666666667*log(sin(2*x)) - 0.0833333333333333*log(4*sin(2*x)^2 + 3)
0.166666666666667*log(sin(2*x)) - 0.0833333333333333*log(4*sin(2*x)^2 + 3)
Степени
[src]
/ 2 \
log\3 + 4*sin (2*x)/ log(sin(2*x))
- -------------------- + -------------
12 6
$$- \frac{\log{\left(4 \sin^{2}{\left(2 x \right)} + 3 \right)}}{12} + \frac{\log{\left(\sin{\left(2 x \right)} \right)}}{6}$$
/ / -2*I*x 2*I*x\ \
/ 2\ |-I*\- e + e / |
| / -2*I*x 2*I*x\ | log|------------------------|
log\3 - \- e + e / / \ 2 /
- ------------------------------ + -----------------------------
12 6
$$\frac{\log{\left(- \frac{i \left(e^{2 i x} - e^{- 2 i x}\right)}{2} \right)}}{6} - \frac{\log{\left(- \left(e^{2 i x} - e^{- 2 i x}\right)^{2} + 3 \right)}}{12}$$
-log(3 - (-exp(-2*i*x) + exp(2*i*x))^2)/12 + log(-i*(-exp(-2*i*x) + exp(2*i*x))/2)/6
Общий знаменатель
[src]
/ 2 \
log\3 + 4*sin (2*x)/ log(sin(2*x))
- -------------------- + -------------
12 6
$$- \frac{\log{\left(4 \sin^{2}{\left(2 x \right)} + 3 \right)}}{12} + \frac{\log{\left(\sin{\left(2 x \right)} \right)}}{6}$$
-log(3 + 4*sin(2*x)^2)/12 + log(sin(2*x))/6
Комбинаторика
[src]
/ 2 \
log\3 + 4*sin (2*x)/ log(sin(2*x))
- -------------------- + -------------
12 6
$$- \frac{\log{\left(4 \sin^{2}{\left(2 x \right)} + 3 \right)}}{12} + \frac{\log{\left(\sin{\left(2 x \right)} \right)}}{6}$$
-log(3 + 4*sin(2*x)^2)/12 + log(sin(2*x))/6