Sr Examen
Simplificación de expresiones/
Descomposición en fracciones simples/
cos(x)^atan(x)*(log(cos(x))/(1+x^2)-atan(x)*sin(x)/cos(x))
¿Cómo vas a descomponer esta cos(x)^atan(x)*(log(cos(x))/(1+x^2)-atan(x)*sin(x)/cos(x)) expresión en fracciones?
Solución
Ha introducido
[src]
atan(x) /log(cos(x)) atan(x)*sin(x)\
cos (x)*|----------- - --------------|
| 2 cos(x) |
\ 1 + x /
$$\left(- \frac{\sin{\left(x \right)} \operatorname{atan}{\left(x \right)}}{\cos{\left(x \right)}} + \frac{\log{\left(\cos{\left(x \right)} \right)}}{x^{2} + 1}\right) \cos^{\operatorname{atan}{\left(x \right)}}{\left(x \right)}$$
cos(x)^atan(x)*(log(cos(x))/(1 + x^2) - atan(x)*sin(x)/cos(x))
Simplificación general
[src]
-1 + atan(x) / / 2\ \
cos (x)*\cos(x)*log(cos(x)) - \1 + x /*atan(x)*sin(x)/
-----------------------------------------------------------------
2
1 + x
$$\frac{\left(- \left(x^{2} + 1\right) \sin{\left(x \right)} \operatorname{atan}{\left(x \right)} + \log{\left(\cos{\left(x \right)} \right)} \cos{\left(x \right)}\right) \cos^{\operatorname{atan}{\left(x \right)} - 1}{\left(x \right)}}{x^{2} + 1}$$
cos(x)^(-1 + atan(x))*(cos(x)*log(cos(x)) - (1 + x^2)*atan(x)*sin(x))/(1 + x^2)
Respuesta numérica
[src]
cos(x)^atan(x)*(log(cos(x))/(1.0 + x^2) - atan(x)*sin(x)/cos(x))
cos(x)^atan(x)*(log(cos(x))/(1.0 + x^2) - atan(x)*sin(x)/cos(x))
Unión de expresiones racionales
[src]
atan(x) / / 2\ \
cos (x)*\cos(x)*log(cos(x)) - \1 + x /*atan(x)*sin(x)/
------------------------------------------------------------
/ 2\
\1 + x /*cos(x)
$$\frac{\left(- \left(x^{2} + 1\right) \sin{\left(x \right)} \operatorname{atan}{\left(x \right)} + \log{\left(\cos{\left(x \right)} \right)} \cos{\left(x \right)}\right) \cos^{\operatorname{atan}{\left(x \right)}}{\left(x \right)}}{\left(x^{2} + 1\right) \cos{\left(x \right)}}$$
cos(x)^atan(x)*(cos(x)*log(cos(x)) - (1 + x^2)*atan(x)*sin(x))/((1 + x^2)*cos(x))
Denominador común
[src]
/ atan(x) atan(x) 2 atan(x) \
-\cos (x)*atan(x)*sin(x) - cos (x)*cos(x)*log(cos(x)) + x *cos (x)*atan(x)*sin(x)/
-----------------------------------------------------------------------------------------------------
2
x *cos(x) + cos(x)
$$- \frac{x^{2} \sin{\left(x \right)} \cos^{\operatorname{atan}{\left(x \right)}}{\left(x \right)} \operatorname{atan}{\left(x \right)} - \log{\left(\cos{\left(x \right)} \right)} \cos{\left(x \right)} \cos^{\operatorname{atan}{\left(x \right)}}{\left(x \right)} + \sin{\left(x \right)} \cos^{\operatorname{atan}{\left(x \right)}}{\left(x \right)} \operatorname{atan}{\left(x \right)}}{x^{2} \cos{\left(x \right)} + \cos{\left(x \right)}}$$
-(cos(x)^atan(x)*atan(x)*sin(x) - cos(x)^atan(x)*cos(x)*log(cos(x)) + x^2*cos(x)^atan(x)*atan(x)*sin(x))/(x^2*cos(x) + cos(x))
Denominador racional
[src]
atan(x) / / 2\ \
cos (x)*\cos(x)*log(cos(x)) - \1 + x /*atan(x)*sin(x)/
------------------------------------------------------------
/ 2\
\1 + x /*cos(x)
$$\frac{\left(- \left(x^{2} + 1\right) \sin{\left(x \right)} \operatorname{atan}{\left(x \right)} + \log{\left(\cos{\left(x \right)} \right)} \cos{\left(x \right)}\right) \cos^{\operatorname{atan}{\left(x \right)}}{\left(x \right)}}{\left(x^{2} + 1\right) \cos{\left(x \right)}}$$
cos(x)^atan(x)*(cos(x)*log(cos(x)) - (1 + x^2)*atan(x)*sin(x))/((1 + x^2)*cos(x))
Potencias
[src]
/ / I*x -I*x\ \
atan(x) | |e e | |
/ I*x -I*x\ |log|---- + -----| / -I*x I*x\ |
|e e | | \ 2 2 / I*\- e + e /*atan(x)|
|---- + -----| *|----------------- + --------------------------|
\ 2 2 / | 2 / I*x -I*x\ |
| 1 + x |e e | |
| 2*|---- + -----| |
\ \ 2 2 / /
$$\left(\frac{i \left(e^{i x} - e^{- i x}\right) \operatorname{atan}{\left(x \right)}}{2 \left(\frac{e^{i x}}{2} + \frac{e^{- i x}}{2}\right)} + \frac{\log{\left(\frac{e^{i x}}{2} + \frac{e^{- i x}}{2} \right)}}{x^{2} + 1}\right) \left(\frac{e^{i x}}{2} + \frac{e^{- i x}}{2}\right)^{\operatorname{atan}{\left(x \right)}}$$
(exp(i*x)/2 + exp(-i*x)/2)^atan(x)*(log(exp(i*x)/2 + exp(-i*x)/2)/(1 + x^2) + i*(-exp(-i*x) + exp(i*x))*atan(x)/(2*(exp(i*x)/2 + exp(-i*x)/2)))
Combinatoria
[src]
atan(x) / 2 \
-cos (x)*\atan(x)*sin(x) - cos(x)*log(cos(x)) + x *atan(x)*sin(x)/
-------------------------------------------------------------------------
/ 2\
\1 + x /*cos(x)
$$- \frac{\left(x^{2} \sin{\left(x \right)} \operatorname{atan}{\left(x \right)} - \log{\left(\cos{\left(x \right)} \right)} \cos{\left(x \right)} + \sin{\left(x \right)} \operatorname{atan}{\left(x \right)}\right) \cos^{\operatorname{atan}{\left(x \right)}}{\left(x \right)}}{\left(x^{2} + 1\right) \cos{\left(x \right)}}$$
-cos(x)^atan(x)*(atan(x)*sin(x) - cos(x)*log(cos(x)) + x^2*atan(x)*sin(x))/((1 + x^2)*cos(x))
Parte trigonométrica
[src]
/ / / pi\\ \
|log|sin|x + --|| |
atan(x)/ pi\ | \ \ 2 // atan(x)*sin(x)|
sin |x + --|*|---------------- - --------------|
\ 2 / | 2 / pi\ |
| 1 + x sin|x + --| |
\ \ 2 / /
$$\left(- \frac{\sin{\left(x \right)} \operatorname{atan}{\left(x \right)}}{\sin{\left(x + \frac{\pi}{2} \right)}} + \frac{\log{\left(\sin{\left(x + \frac{\pi}{2} \right)} \right)}}{x^{2} + 1}\right) \sin^{\operatorname{atan}{\left(x \right)}}{\left(x + \frac{\pi}{2} \right)}$$
/ / 2/x\\ \
| |-1 + cot |-|| |
| | \2/| |
atan(x) |log|------------| |
/ 2/x\\ | | 2/x\ | /x\|
|-1 + cot |-|| | |1 + cot |-| | 2*atan(x)*cot|-||
| \2/| | \ \2/ / \2/|
|------------| *|----------------- - ----------------|
| 2/x\ | | 2 2/x\ |
|1 + cot |-| | | 1 + x -1 + cot |-| |
\ \2/ / \ \2/ /
$$\left(\frac{\cot^{2}{\left(\frac{x}{2} \right)} - 1}{\cot^{2}{\left(\frac{x}{2} \right)} + 1}\right)^{\operatorname{atan}{\left(x \right)}} \left(- \frac{2 \cot{\left(\frac{x}{2} \right)} \operatorname{atan}{\left(x \right)}}{\cot^{2}{\left(\frac{x}{2} \right)} - 1} + \frac{\log{\left(\frac{\cot^{2}{\left(\frac{x}{2} \right)} - 1}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} \right)}}{x^{2} + 1}\right)$$
/ / 1 \ \
atan(x) |log|------| |
/ 1 \ | \sec(x)/ atan(x)*sec(x)|
|------| *|----------- - --------------|
\sec(x)/ | 2 / pi\ |
| 1 + x sec|x - --| |
\ \ 2 / /
$$\left(- \frac{\operatorname{atan}{\left(x \right)} \sec{\left(x \right)}}{\sec{\left(x - \frac{\pi}{2} \right)}} + \frac{\log{\left(\frac{1}{\sec{\left(x \right)}} \right)}}{x^{2} + 1}\right) \left(\frac{1}{\sec{\left(x \right)}}\right)^{\operatorname{atan}{\left(x \right)}}$$
/ / 1 \ \
atan(x) |log|------| |
/ 1 \ | \sec(x)/ atan(x)*sec(x)|
|------| *|----------- - --------------|
\sec(x)/ | 2 csc(x) |
\ 1 + x /
$$\left(- \frac{\operatorname{atan}{\left(x \right)} \sec{\left(x \right)}}{\csc{\left(x \right)}} + \frac{\log{\left(\frac{1}{\sec{\left(x \right)}} \right)}}{x^{2} + 1}\right) \left(\frac{1}{\sec{\left(x \right)}}\right)^{\operatorname{atan}{\left(x \right)}}$$
/ / 2/x\\ \
| |1 - tan |-|| |
| | \2/| |
atan(x) |log|-----------| |
/ 2/x\\ | | 2/x\| /x\|
|1 - tan |-|| | |1 + tan |-|| 2*atan(x)*tan|-||
| \2/| | \ \2// \2/|
|-----------| *|---------------- - ----------------|
| 2/x\| | 2 2/x\ |
|1 + tan |-|| | 1 + x 1 - tan |-| |
\ \2// \ \2/ /
$$\left(\frac{1 - \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1}\right)^{\operatorname{atan}{\left(x \right)}} \left(\frac{\log{\left(\frac{1 - \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} \right)}}{x^{2} + 1} - \frac{2 \tan{\left(\frac{x}{2} \right)} \operatorname{atan}{\left(x \right)}}{1 - \tan^{2}{\left(\frac{x}{2} \right)}}\right)$$
atan(x) /log(cos(x)) \
cos (x)*|----------- - atan(x)*tan(x)|
| 2 |
\ 1 + x /
$$\left(- \tan{\left(x \right)} \operatorname{atan}{\left(x \right)} + \frac{\log{\left(\cos{\left(x \right)} \right)}}{x^{2} + 1}\right) \cos^{\operatorname{atan}{\left(x \right)}}{\left(x \right)}$$
/ / / pi\\ \
|log|sin|x + --|| 2 |
atan(x)/ pi\ | \ \ 2 // 2*sin (x)*atan(x)|
sin |x + --|*|---------------- - -----------------|
\ 2 / | 2 sin(2*x) |
\ 1 + x /
$$\left(- \frac{2 \sin^{2}{\left(x \right)} \operatorname{atan}{\left(x \right)}}{\sin{\left(2 x \right)}} + \frac{\log{\left(\sin{\left(x + \frac{\pi}{2} \right)} \right)}}{x^{2} + 1}\right) \sin^{\operatorname{atan}{\left(x \right)}}{\left(x + \frac{\pi}{2} \right)}$$
/ / pi\\
| atan(x)*cos|x - --||
atan(x) |log(cos(x)) \ 2 /|
cos (x)*|----------- - -------------------|
| 2 cos(x) |
\ 1 + x /
$$\left(- \frac{\cos{\left(x - \frac{\pi}{2} \right)} \operatorname{atan}{\left(x \right)}}{\cos{\left(x \right)}} + \frac{\log{\left(\cos{\left(x \right)} \right)}}{x^{2} + 1}\right) \cos^{\operatorname{atan}{\left(x \right)}}{\left(x \right)}$$
/ / 2/x\\ \
| |-1 + cot |-|| |
| | \2/| |
atan(x) |log|------------| |
/ 2/x\\ | | 2/x\ | |
|-1 + cot |-|| | |1 + cot |-| | |
| \2/| | \ \2/ / atan(x)|
|------------| *|----------------- - -------|
| 2/x\ | | 2 cot(x)|
|1 + cot |-| | \ 1 + x /
\ \2/ /
$$\left(\frac{\cot^{2}{\left(\frac{x}{2} \right)} - 1}{\cot^{2}{\left(\frac{x}{2} \right)} + 1}\right)^{\operatorname{atan}{\left(x \right)}} \left(- \frac{\operatorname{atan}{\left(x \right)}}{\cot{\left(x \right)}} + \frac{\log{\left(\frac{\cot^{2}{\left(\frac{x}{2} \right)} - 1}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} \right)}}{x^{2} + 1}\right)$$
/ / 2/x\\ \
| |1 - tan |-|| |
| | \2/| |
atan(x) |log|-----------| |
/ 2/x\\ | | 2/x\| |
|1 - tan |-|| | |1 + tan |-|| |
| \2/| | \ \2// |
|-----------| *|---------------- - atan(x)*tan(x)|
| 2/x\| | 2 |
|1 + tan |-|| \ 1 + x /
\ \2//
$$\left(\frac{1 - \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1}\right)^{\operatorname{atan}{\left(x \right)}} \left(- \tan{\left(x \right)} \operatorname{atan}{\left(x \right)} + \frac{\log{\left(\frac{1 - \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} \right)}}{x^{2} + 1}\right)$$
/ / 1 \ \
|log|-----------| |
| | /pi \| /pi \|
atan(x) | |csc|-- - x|| atan(x)*csc|-- - x||
/ 1 \ | \ \2 // \2 /|
|-----------| *|---------------- - -------------------|
| /pi \| | 2 csc(x) |
|csc|-- - x|| \ 1 + x /
\ \2 //
$$\left(- \frac{\operatorname{atan}{\left(x \right)} \csc{\left(- x + \frac{\pi}{2} \right)}}{\csc{\left(x \right)}} + \frac{\log{\left(\frac{1}{\csc{\left(- x + \frac{\pi}{2} \right)}} \right)}}{x^{2} + 1}\right) \left(\frac{1}{\csc{\left(- x + \frac{\pi}{2} \right)}}\right)^{\operatorname{atan}{\left(x \right)}}$$
(1/csc(pi/2 - x))^atan(x)*(log(1/csc(pi/2 - x))/(1 + x^2) - atan(x)*csc(pi/2 - x)/csc(x))