Sr Examen
Simplificación de expresiones/
Descomposición en fracciones simples/
sin(2*x)^((1/x)^2)*(-2*log(sin(2*x))/x^3+2*cos(2*x)/(x^2*sin(2*x)))
¿Cómo vas a descomponer esta sin(2*x)^((1/x)^2)*(-2*log(sin(2*x))/x^3+2*cos(2*x)/(x^2*sin(2*x))) expresión en fracciones?
Solución
Ha introducido
[src]
/ 2\
|/1\ |
||-| |
\\x/ / /-2*log(sin(2*x)) 2*cos(2*x)\
(sin(2*x)) *|---------------- + -----------|
| 3 2 |
\ x x *sin(2*x)/
$$\left(\frac{2 \cos{\left(2 x \right)}}{x^{2} \sin{\left(2 x \right)}} + \frac{\left(-1\right) 2 \log{\left(\sin{\left(2 x \right)} \right)}}{x^{3}}\right) \sin^{\left(\frac{1}{x}\right)^{2}}{\left(2 x \right)}$$
sin(2*x)^((1/x)^2)*((-2*log(sin(2*x)))/x^3 + (2*cos(2*x))/((x^2*sin(2*x))))
Simplificación general
[src]
2
1 - x
------
2
x
2*(sin(2*x)) *(x*cos(2*x) - log(sin(2*x))*sin(2*x))
--------------------------------------------------------
3
x
$$\frac{2 \left(x \cos{\left(2 x \right)} - \log{\left(\sin{\left(2 x \right)} \right)} \sin{\left(2 x \right)}\right) \sin^{\frac{1 - x^{2}}{x^{2}}}{\left(2 x \right)}}{x^{3}}$$
2*sin(2*x)^((1 - x^2)/x^2)*(x*cos(2*x) - log(sin(2*x))*sin(2*x))/x^3
Respuesta numérica
[src]
sin(2*x)^(x^(-2))*(-2.0*log(sin(2*x))/x^3 + 2.0*cos(2*x)/(x^2*sin(2*x)))
sin(2*x)^(x^(-2))*(-2.0*log(sin(2*x))/x^3 + 2.0*cos(2*x)/(x^2*sin(2*x)))
Denominador común
[src]
1 1
-- --
2 2
x x
- 2*(sin(2*x)) *log(sin(2*x))*sin(2*x) + 2*x*(sin(2*x)) *cos(2*x)
-------------------------------------------------------------------
3
x *sin(2*x)
$$\frac{2 x \sin^{\frac{1}{x^{2}}}{\left(2 x \right)} \cos{\left(2 x \right)} - 2 \log{\left(\sin{\left(2 x \right)} \right)} \sin{\left(2 x \right)} \sin^{\frac{1}{x^{2}}}{\left(2 x \right)}}{x^{3} \sin{\left(2 x \right)}}$$
(-2*sin(2*x)^(x^(-2))*log(sin(2*x))*sin(2*x) + 2*x*sin(2*x)^(x^(-2))*cos(2*x))/(x^3*sin(2*x))
Denominador racional
[src]
1 1
-1 + -- -1 + --
2 2
3 x 2 x
2*x *(sin(2*x)) *cos(2*x) - 2*x *(sin(2*x)) *log(sin(2*x))*sin(2*x)
-------------------------------------------------------------------------------
5
x
$$\frac{2 x^{3} \sin^{-1 + \frac{1}{x^{2}}}{\left(2 x \right)} \cos{\left(2 x \right)} - 2 x^{2} \log{\left(\sin{\left(2 x \right)} \right)} \sin{\left(2 x \right)} \sin^{-1 + \frac{1}{x^{2}}}{\left(2 x \right)}}{x^{5}}$$
(2*x^3*sin(2*x)^(-1 + x^(-2))*cos(2*x) - 2*x^2*sin(2*x)^(-1 + x^(-2))*log(sin(2*x))*sin(2*x))/x^5
Combinatoria
[src]
1
--
2
x
2*(sin(2*x)) *(x*cos(2*x) - log(sin(2*x))*sin(2*x))
----------------------------------------------------
3
x *sin(2*x)
$$\frac{2 \left(x \cos{\left(2 x \right)} - \log{\left(\sin{\left(2 x \right)} \right)} \sin{\left(2 x \right)}\right) \sin^{\frac{1}{x^{2}}}{\left(2 x \right)}}{x^{3} \sin{\left(2 x \right)}}$$
2*sin(2*x)^(x^(-2))*(x*cos(2*x) - log(sin(2*x))*sin(2*x))/(x^3*sin(2*x))
Compilar la expresión
[src]
/ 2\
|/1\ |
||-| |
\\x/ / / 2*log(sin(2*x)) 2*cos(2*x)\
(sin(2*x)) *|- --------------- + -----------|
| 3 2 |
\ x x *sin(2*x)/
$$\left(\frac{2 \cos{\left(2 x \right)}}{x^{2} \sin{\left(2 x \right)}} - \frac{2 \log{\left(\sin{\left(2 x \right)} \right)}}{x^{3}}\right) \sin^{\left(\frac{1}{x}\right)^{2}}{\left(2 x \right)}$$
sin(2*x)^((1/x)^2)*(-2*log(sin(2*x))/x^3 + 2*cos(2*x)/(x^2*sin(2*x)))
Unión de expresiones racionales
[src]
1
--
2
x
2*(sin(2*x)) *(x*cos(2*x) - log(sin(2*x))*sin(2*x))
----------------------------------------------------
3
x *sin(2*x)
$$\frac{2 \left(x \cos{\left(2 x \right)} - \log{\left(\sin{\left(2 x \right)} \right)} \sin{\left(2 x \right)}\right) \sin^{\frac{1}{x^{2}}}{\left(2 x \right)}}{x^{3} \sin{\left(2 x \right)}}$$
2*sin(2*x)^(x^(-2))*(x*cos(2*x) - log(sin(2*x))*sin(2*x))/(x^3*sin(2*x))
Parte trigonométrica
[src]
1
--
2
x / 2*log(sin(2*x)) 2*cos(2*x)\
(sin(2*x)) *|- --------------- + -----------|
| 3 2 |
\ x x *sin(2*x)/
$$\left(\frac{2 \cos{\left(2 x \right)}}{x^{2} \sin{\left(2 x \right)}} - \frac{2 \log{\left(\sin{\left(2 x \right)} \right)}}{x^{3}}\right) \sin^{\frac{1}{x^{2}}}{\left(2 x \right)}$$
1
-- / / 2*tan(x) \ \
2 | 2*log|-----------| |
x | | 2 | 2 |
/ 2*tan(x) \ | \1 + tan (x)/ 1 - tan (x)|
|-----------| *|- ------------------ + -----------|
| 2 | | 3 2 |
\1 + tan (x)/ \ x x *tan(x) /
$$\left(\frac{2 \tan{\left(x \right)}}{\tan^{2}{\left(x \right)} + 1}\right)^{\frac{1}{x^{2}}} \left(\frac{1 - \tan^{2}{\left(x \right)}}{x^{2} \tan{\left(x \right)}} - \frac{2 \log{\left(\frac{2 \tan{\left(x \right)}}{\tan^{2}{\left(x \right)} + 1} \right)}}{x^{3}}\right)$$
1
-- / / 2*tan(x) \\
2 | 1 2*log|-----------||
x |------ - tan(x) | 2 ||
/ 2*tan(x) \ |tan(x) \1 + tan (x)/|
|-----------| *|--------------- - ------------------|
| 2 | | 2 3 |
\1 + tan (x)/ \ x x /
$$\left(\frac{2 \tan{\left(x \right)}}{\tan^{2}{\left(x \right)} + 1}\right)^{\frac{1}{x^{2}}} \left(\frac{- \tan{\left(x \right)} + \frac{1}{\tan{\left(x \right)}}}{x^{2}} - \frac{2 \log{\left(\frac{2 \tan{\left(x \right)}}{\tan^{2}{\left(x \right)} + 1} \right)}}{x^{3}}\right)$$
/ /pi \ \
| csc|-- - x| |
1 | csc(x) \2 / |
-- |----------- - ----------- |
2 | /pi \ csc(x) / 1 \|
x |csc|-- - x| 2*log|--------||
/ 1 \ | \2 / \csc(2*x)/|
|--------| *|------------------------- - ---------------|
\csc(2*x)/ | 2 3 |
\ x x /
$$\left(\frac{\frac{\csc{\left(x \right)}}{\csc{\left(- x + \frac{\pi}{2} \right)}} - \frac{\csc{\left(- x + \frac{\pi}{2} \right)}}{\csc{\left(x \right)}}}{x^{2}} - \frac{2 \log{\left(\frac{1}{\csc{\left(2 x \right)}} \right)}}{x^{3}}\right) \left(\frac{1}{\csc{\left(2 x \right)}}\right)^{\frac{1}{x^{2}}}$$
1
--
2 / / / pi\\ \
x | 2*log|cos|2*x - --|| |
/ / pi\\ | \ \ 2 // 2*cos(2*x) |
|cos|2*x - --|| *|- -------------------- + ----------------|
\ \ 2 // | 3 2 / pi\|
| x x *cos|2*x - --||
\ \ 2 //
$$\left(\frac{2 \cos{\left(2 x \right)}}{x^{2} \cos{\left(2 x - \frac{\pi}{2} \right)}} - \frac{2 \log{\left(\cos{\left(2 x - \frac{\pi}{2} \right)} \right)}}{x^{3}}\right) \cos^{\frac{1}{x^{2}}}{\left(2 x - \frac{\pi}{2} \right)}$$
/ 2 \
1 | sin(2*x) 2*sin (x) |
-- |--------- - --------- |
2 | 2 sin(2*x) |
x |2*sin (x) 2*log(sin(2*x))|
(sin(2*x)) *|--------------------- - ---------------|
| 2 3 |
\ x x /
$$\left(\frac{- \frac{2 \sin^{2}{\left(x \right)}}{\sin{\left(2 x \right)}} + \frac{\sin{\left(2 x \right)}}{2 \sin^{2}{\left(x \right)}}}{x^{2}} - \frac{2 \log{\left(\sin{\left(2 x \right)} \right)}}{x^{3}}\right) \sin^{\frac{1}{x^{2}}}{\left(2 x \right)}$$
1
-- / / 2*cot(x) \\
2 | 1 2*log|-----------||
x |- ------ + cot(x) | 2 ||
/ 2*cot(x) \ | cot(x) \1 + cot (x)/|
|-----------| *|----------------- - ------------------|
| 2 | | 2 3 |
\1 + cot (x)/ \ x x /
$$\left(\frac{2 \cot{\left(x \right)}}{\cot^{2}{\left(x \right)} + 1}\right)^{\frac{1}{x^{2}}} \left(\frac{\cot{\left(x \right)} - \frac{1}{\cot{\left(x \right)}}}{x^{2}} - \frac{2 \log{\left(\frac{2 \cot{\left(x \right)}}{\cot^{2}{\left(x \right)} + 1} \right)}}{x^{3}}\right)$$
1
-- / 1 \
2 |------ - tan(x) |
x |tan(x) 2*log(sin(2*x))|
(sin(2*x)) *|--------------- - ---------------|
| 2 3 |
\ x x /
$$\left(\frac{- \tan{\left(x \right)} + \frac{1}{\tan{\left(x \right)}}}{x^{2}} - \frac{2 \log{\left(\sin{\left(2 x \right)} \right)}}{x^{3}}\right) \sin^{\frac{1}{x^{2}}}{\left(2 x \right)}$$
1
--
2
x / 2*log(sin(2*x)) sin(4*x) \
(sin(2*x)) *|- --------------- + ------------|
| 3 2 2 |
\ x x *sin (2*x)/
$$\left(\frac{\sin{\left(4 x \right)}}{x^{2} \sin^{2}{\left(2 x \right)}} - \frac{2 \log{\left(\sin{\left(2 x \right)} \right)}}{x^{3}}\right) \sin^{\frac{1}{x^{2}}}{\left(2 x \right)}$$
1 / / 1 \ \
-- | 2*log|-------------| |
2 | | / pi\| / pi\|
x | |sec|2*x - --|| 2*sec|2*x - --||
/ 1 \ | \ \ 2 // \ 2 /|
|-------------| *|- -------------------- + ---------------|
| / pi\| | 3 2 |
|sec|2*x - --|| \ x x *sec(2*x) /
\ \ 2 //
$$\left(\frac{2 \sec{\left(2 x - \frac{\pi}{2} \right)}}{x^{2} \sec{\left(2 x \right)}} - \frac{2 \log{\left(\frac{1}{\sec{\left(2 x - \frac{\pi}{2} \right)}} \right)}}{x^{3}}\right) \left(\frac{1}{\sec{\left(2 x - \frac{\pi}{2} \right)}}\right)^{\frac{1}{x^{2}}}$$
1
--
2 / / 1 \ \
x | 2*log|--------| |
/ 1 \ | \csc(2*x)/ 2*csc(2*x) |
|--------| *|- --------------- + ----------------|
\csc(2*x)/ | 3 2 /pi \|
| x x *csc|-- - 2*x||
\ \2 //
$$\left(\frac{2 \csc{\left(2 x \right)}}{x^{2} \csc{\left(- 2 x + \frac{\pi}{2} \right)}} - \frac{2 \log{\left(\frac{1}{\csc{\left(2 x \right)}} \right)}}{x^{3}}\right) \left(\frac{1}{\csc{\left(2 x \right)}}\right)^{\frac{1}{x^{2}}}$$
1
--
2
x / 2*log(sin(2*x)) 2*cot(2*x)\
(sin(2*x)) *|- --------------- + ----------|
| 3 2 |
\ x x /
$$\left(\frac{2 \cot{\left(2 x \right)}}{x^{2}} - \frac{2 \log{\left(\sin{\left(2 x \right)} \right)}}{x^{3}}\right) \sin^{\frac{1}{x^{2}}}{\left(2 x \right)}$$
/ / pi\ \
| cos|x - --| |
1 | cos(x) \ 2 / |
-- |----------- - ----------- |
2 | / pi\ cos(x) / / pi\\|
x |cos|x - --| 2*log|cos|2*x - --|||
/ / pi\\ | \ 2 / \ \ 2 //|
|cos|2*x - --|| *|------------------------- - --------------------|
\ \ 2 // | 2 3 |
\ x x /
$$\left(\frac{\frac{\cos{\left(x \right)}}{\cos{\left(x - \frac{\pi}{2} \right)}} - \frac{\cos{\left(x - \frac{\pi}{2} \right)}}{\cos{\left(x \right)}}}{x^{2}} - \frac{2 \log{\left(\cos{\left(2 x - \frac{\pi}{2} \right)} \right)}}{x^{3}}\right) \cos^{\frac{1}{x^{2}}}{\left(2 x - \frac{\pi}{2} \right)}$$
/ 2\
|/1\ |
||-| |
\\x/ / / 2*log(sin(2*x)) 2 \
(sin(2*x)) *|- --------------- + -----------|
| 3 2 |
\ x x *tan(2*x)/
$$\left(\frac{2}{x^{2} \tan{\left(2 x \right)}} - \frac{2 \log{\left(\sin{\left(2 x \right)} \right)}}{x^{3}}\right) \sin^{\left(\frac{1}{x}\right)^{2}}{\left(2 x \right)}$$
1
--
2
x / 2*log(sin(2*x)) 2 \
(sin(2*x)) *|- --------------- + -----------|
| 3 2 |
\ x x *tan(2*x)/
$$\left(\frac{2}{x^{2} \tan{\left(2 x \right)}} - \frac{2 \log{\left(\sin{\left(2 x \right)} \right)}}{x^{3}}\right) \sin^{\frac{1}{x^{2}}}{\left(2 x \right)}$$
1
-- / / 2*cot(x) \ \
2 | 2*log|-----------| |
x | | 2 | 2 |
/ 2*cot(x) \ | \1 + cot (x)/ -1 + cot (x)|
|-----------| *|- ------------------ + ------------|
| 2 | | 3 2 |
\1 + cot (x)/ \ x x *cot(x) /
$$\left(\frac{2 \cot{\left(x \right)}}{\cot^{2}{\left(x \right)} + 1}\right)^{\frac{1}{x^{2}}} \left(\frac{\cot^{2}{\left(x \right)} - 1}{x^{2} \cot{\left(x \right)}} - \frac{2 \log{\left(\frac{2 \cot{\left(x \right)}}{\cot^{2}{\left(x \right)} + 1} \right)}}{x^{3}}\right)$$
1
--
2 / / 1 \ \
x | 2*log|--------| |
/ 1 \ | \csc(2*x)/ 2*csc(2*x)|
|--------| *|- --------------- + -----------|
\csc(2*x)/ | 3 2 |
\ x x *sec(2*x)/
$$\left(\frac{2 \csc{\left(2 x \right)}}{x^{2} \sec{\left(2 x \right)}} - \frac{2 \log{\left(\frac{1}{\csc{\left(2 x \right)}} \right)}}{x^{3}}\right) \left(\frac{1}{\csc{\left(2 x \right)}}\right)^{\frac{1}{x^{2}}}$$
1
--
2
x /-tan(x) + cot(x) 2*log(sin(2*x))\
(sin(2*x)) *|---------------- - ---------------|
| 2 3 |
\ x x /
$$\left(\frac{- \tan{\left(x \right)} + \cot{\left(x \right)}}{x^{2}} - \frac{2 \log{\left(\sin{\left(2 x \right)} \right)}}{x^{3}}\right) \sin^{\frac{1}{x^{2}}}{\left(2 x \right)}$$
1
-- / /pi \\
2 | 2*sin|-- + 2*x||
x | 2*log(sin(2*x)) \2 /|
(sin(2*x)) *|- --------------- + ---------------|
| 3 2 |
\ x x *sin(2*x) /
$$\left(\frac{2 \sin{\left(2 x + \frac{\pi}{2} \right)}}{x^{2} \sin{\left(2 x \right)}} - \frac{2 \log{\left(\sin{\left(2 x \right)} \right)}}{x^{3}}\right) \sin^{\frac{1}{x^{2}}}{\left(2 x \right)}$$
/ / pi\ \
|sec|x - --| |
1 | \ 2 / sec(x) / 1 \|
-- |----------- - ----------- 2*log|-------------||
2 | sec(x) / pi\ | / pi\||
x | sec|x - --| |sec|2*x - --|||
/ 1 \ | \ 2 / \ \ 2 //|
|-------------| *|------------------------- - --------------------|
| / pi\| | 2 3 |
|sec|2*x - --|| \ x x /
\ \ 2 //
$$\left(\frac{- \frac{\sec{\left(x \right)}}{\sec{\left(x - \frac{\pi}{2} \right)}} + \frac{\sec{\left(x - \frac{\pi}{2} \right)}}{\sec{\left(x \right)}}}{x^{2}} - \frac{2 \log{\left(\frac{1}{\sec{\left(2 x - \frac{\pi}{2} \right)}} \right)}}{x^{3}}\right) \left(\frac{1}{\sec{\left(2 x - \frac{\pi}{2} \right)}}\right)^{\frac{1}{x^{2}}}$$
1
-- / / 2*tan(x) \ \
2 | 2*log|-----------| |
x | | 2 | |
/ 2*tan(x) \ | \1 + tan (x)/ 2 |
|-----------| *|- ------------------ + -----------|
| 2 | | 3 2 |
\1 + tan (x)/ \ x x *tan(2*x)/
$$\left(\frac{2 \tan{\left(x \right)}}{\tan^{2}{\left(x \right)} + 1}\right)^{\frac{1}{x^{2}}} \left(\frac{2}{x^{2} \tan{\left(2 x \right)}} - \frac{2 \log{\left(\frac{2 \tan{\left(x \right)}}{\tan^{2}{\left(x \right)} + 1} \right)}}{x^{3}}\right)$$
1
-- / / 2*cot(x) \ \
2 | 2*log|-----------| |
x | | 2 | |
/ 2*cot(x) \ | \1 + cot (x)/ 2*cot(2*x)|
|-----------| *|- ------------------ + ----------|
| 2 | | 3 2 |
\1 + cot (x)/ \ x x /
$$\left(\frac{2 \cot{\left(x \right)}}{\cot^{2}{\left(x \right)} + 1}\right)^{\frac{1}{x^{2}}} \left(\frac{2 \cot{\left(2 x \right)}}{x^{2}} - \frac{2 \log{\left(\frac{2 \cot{\left(x \right)}}{\cot^{2}{\left(x \right)} + 1} \right)}}{x^{3}}\right)$$
(2*cot(x)/(1 + cot(x)^2))^(x^(-2))*(-2*log(2*cot(x)/(1 + cot(x)^2))/x^3 + 2*cot(2*x)/x^2)
Potencias
[src]
1
--
2
x / 2*log(sin(2*x)) 2*cos(2*x)\
(sin(2*x)) *|- --------------- + -----------|
| 3 2 |
\ x x *sin(2*x)/
$$\left(\frac{2 \cos{\left(2 x \right)}}{x^{2} \sin{\left(2 x \right)}} - \frac{2 \log{\left(\sin{\left(2 x \right)} \right)}}{x^{3}}\right) \sin^{\frac{1}{x^{2}}}{\left(2 x \right)}$$
1
--
2 / / / -2*I*x 2*I*x\ \ \
x | |-I*\- e + e / | |
/ / -2*I*x 2*I*x\ \ | 2*log|------------------------| / -2*I*x 2*I*x\|
|-I*\- e + e / | | \ 2 / 2*I*\e + e /|
|------------------------| *|- ------------------------------- + -----------------------|
\ 2 / | 3 2 / -2*I*x 2*I*x\|
\ x x *\- e + e //
$$\left(- \frac{i \left(e^{2 i x} - e^{- 2 i x}\right)}{2}\right)^{\frac{1}{x^{2}}} \left(\frac{2 i \left(e^{2 i x} + e^{- 2 i x}\right)}{x^{2} \left(e^{2 i x} - e^{- 2 i x}\right)} - \frac{2 \log{\left(- \frac{i \left(e^{2 i x} - e^{- 2 i x}\right)}{2} \right)}}{x^{3}}\right)$$
(-i*(-exp(-2*i*x) + exp(2*i*x))/2)^(x^(-2))*(-2*log(-i*(-exp(-2*i*x) + exp(2*i*x))/2)/x^3 + 2*i*(exp(-2*i*x) + exp(2*i*x))/(x^2*(-exp(-2*i*x) + exp(2*i*x))))