Sr Examen
Simplificación de expresiones/
Descomposición en fracciones simples/
tan(x)^atan(x)*(log(tan(x))/(1+x^2)+(1+tan(x)^2)*atan(x)/tan(x))
¿Cómo vas a descomponer esta tan(x)^atan(x)*(log(tan(x))/(1+x^2)+(1+tan(x)^2)*atan(x)/tan(x)) expresión en fracciones?
Solución
Ha introducido
[src]
/ / 2 \ \
atan(x) |log(tan(x)) \1 + tan (x)/*atan(x)|
tan (x)*|----------- + ---------------------|
| 2 tan(x) |
\ 1 + x /
$$\left(\frac{\left(\tan^{2}{\left(x \right)} + 1\right) \operatorname{atan}{\left(x \right)}}{\tan{\left(x \right)}} + \frac{\log{\left(\tan{\left(x \right)} \right)}}{x^{2} + 1}\right) \tan^{\operatorname{atan}{\left(x \right)}}{\left(x \right)}$$
tan(x)^atan(x)*(log(tan(x))/(1 + x^2) + ((1 + tan(x)^2)*atan(x))/tan(x))
Simplificación general
[src]
/ / 2\ \
-1 + atan(x) | \1 + x /*atan(x)|
tan (x)*|log(tan(x))*tan(x) + ----------------|
| 2 |
\ cos (x) /
----------------------------------------------------------
2
1 + x
$$\frac{\left(\frac{\left(x^{2} + 1\right) \operatorname{atan}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + \log{\left(\tan{\left(x \right)} \right)} \tan{\left(x \right)}\right) \tan^{\operatorname{atan}{\left(x \right)} - 1}{\left(x \right)}}{x^{2} + 1}$$
tan(x)^(-1 + atan(x))*(log(tan(x))*tan(x) + (1 + x^2)*atan(x)/cos(x)^2)/(1 + x^2)
Respuesta numérica
[src]
tan(x)^atan(x)*(log(tan(x))/(1.0 + x^2) + (1.0 + tan(x)^2)*atan(x)/tan(x))
tan(x)^atan(x)*(log(tan(x))/(1.0 + x^2) + (1.0 + tan(x)^2)*atan(x)/tan(x))
Unión de expresiones racionales
[src]
atan(x) / / 2\ / 2 \ \
tan (x)*\log(tan(x))*tan(x) + \1 + x /*\1 + tan (x)/*atan(x)/
-------------------------------------------------------------------
/ 2\
\1 + x /*tan(x)
$$\frac{\left(\left(x^{2} + 1\right) \left(\tan^{2}{\left(x \right)} + 1\right) \operatorname{atan}{\left(x \right)} + \log{\left(\tan{\left(x \right)} \right)} \tan{\left(x \right)}\right) \tan^{\operatorname{atan}{\left(x \right)}}{\left(x \right)}}{\left(x^{2} + 1\right) \tan{\left(x \right)}}$$
tan(x)^atan(x)*(log(tan(x))*tan(x) + (1 + x^2)*(1 + tan(x)^2)*atan(x))/((1 + x^2)*tan(x))
Combinatoria
[src]
atan(x) / 2 2 2 2 \
tan (x)*\x *atan(x) + tan (x)*atan(x) + log(tan(x))*tan(x) + x *tan (x)*atan(x) + atan(x)/
------------------------------------------------------------------------------------------------
/ 2\
\1 + x /*tan(x)
$$\frac{\left(x^{2} \tan^{2}{\left(x \right)} \operatorname{atan}{\left(x \right)} + x^{2} \operatorname{atan}{\left(x \right)} + \log{\left(\tan{\left(x \right)} \right)} \tan{\left(x \right)} + \tan^{2}{\left(x \right)} \operatorname{atan}{\left(x \right)} + \operatorname{atan}{\left(x \right)}\right) \tan^{\operatorname{atan}{\left(x \right)}}{\left(x \right)}}{\left(x^{2} + 1\right) \tan{\left(x \right)}}$$
tan(x)^atan(x)*(x^2*atan(x) + tan(x)^2*atan(x) + log(tan(x))*tan(x) + x^2*tan(x)^2*atan(x) + atan(x))/((1 + x^2)*tan(x))
Potencias
[src]
/ / 2\ \
| | / I*x -I*x\ | |
| / / I*x -I*x\\ | \- e + e / | / I*x -I*x\ |
| |I*\- e + e /| I*|1 - -----------------|*\e + e /*atan(x)|
atan(x) |log|------------------| | 2 | |
/ / I*x -I*x\\ | | I*x -I*x | | / I*x -I*x\ | |
|I*\- e + e /| | \ e + e / \ \e + e / / |
|------------------| *|----------------------- - ------------------------------------------------|
| I*x -I*x | | 2 I*x -I*x |
\ e + e / \ 1 + x - e + e /
$$\left(\frac{i \left(- e^{i x} + e^{- i x}\right)}{e^{i x} + e^{- i x}}\right)^{\operatorname{atan}{\left(x \right)}} \left(- \frac{i \left(- \frac{\left(- e^{i x} + e^{- i x}\right)^{2}}{\left(e^{i x} + e^{- i x}\right)^{2}} + 1\right) \left(e^{i x} + e^{- i x}\right) \operatorname{atan}{\left(x \right)}}{- e^{i x} + e^{- i x}} + \frac{\log{\left(\frac{i \left(- e^{i x} + e^{- i x}\right)}{e^{i x} + e^{- i x}} \right)}}{x^{2} + 1}\right)$$
(i*(-exp(i*x) + exp(-i*x))/(exp(i*x) + exp(-i*x)))^atan(x)*(log(i*(-exp(i*x) + exp(-i*x))/(exp(i*x) + exp(-i*x)))/(1 + x^2) - i*(1 - (-exp(i*x) + exp(-i*x))^2/(exp(i*x) + exp(-i*x))^2)*(exp(i*x) + exp(-i*x))*atan(x)/(-exp(i*x) + exp(-i*x)))
Parte trigonométrica
[src]
atan(x) /log(tan(x)) \
tan (x)*|----------- + 2*atan(x)*csc(2*x)|
| 2 |
\ 1 + x /
$$\left(2 \operatorname{atan}{\left(x \right)} \csc{\left(2 x \right)} + \frac{\log{\left(\tan{\left(x \right)} \right)}}{x^{2} + 1}\right) \tan^{\operatorname{atan}{\left(x \right)}}{\left(x \right)}$$
/ / 2 \ \
| | sec (x)| |
| /sec(x)\ |1 + -------|*atan(x)*csc(x)|
atan(x) |log|------| | 2 | |
/sec(x)\ | \csc(x)/ \ csc (x)/ |
|------| *|----------- + ----------------------------|
\csc(x)/ | 2 sec(x) |
\ 1 + x /
$$\left(\frac{\sec{\left(x \right)}}{\csc{\left(x \right)}}\right)^{\operatorname{atan}{\left(x \right)}} \left(\frac{\left(1 + \frac{\sec^{2}{\left(x \right)}}{\csc^{2}{\left(x \right)}}\right) \operatorname{atan}{\left(x \right)} \csc{\left(x \right)}}{\sec{\left(x \right)}} + \frac{\log{\left(\frac{\sec{\left(x \right)}}{\csc{\left(x \right)}} \right)}}{x^{2} + 1}\right)$$
/ / / pi\\ \
atan(x) | |cos|x - --|| |
/ / pi\\ | | \ 2 /| |
|cos|x - --|| |log|-----------| |
| \ 2 /| | \ cos(x) / 2*atan(x) |
|-----------| *|---------------- + -------------|
\ cos(x) / | 2 / pi\|
| 1 + x cos|2*x - --||
\ \ 2 //
$$\left(\frac{\cos{\left(x - \frac{\pi}{2} \right)}}{\cos{\left(x \right)}}\right)^{\operatorname{atan}{\left(x \right)}} \left(\frac{2 \operatorname{atan}{\left(x \right)}}{\cos{\left(2 x - \frac{\pi}{2} \right)}} + \frac{\log{\left(\frac{\cos{\left(x - \frac{\pi}{2} \right)}}{\cos{\left(x \right)}} \right)}}{x^{2} + 1}\right)$$
/ / 4 \ \
| / 2 \ | 4*sin (x)| |
atan(x) | |2*sin (x)| |1 + ---------|*atan(x)*sin(2*x)|
/ 2 \ |log|---------| | 2 | |
|2*sin (x)| | \ sin(2*x)/ \ sin (2*x)/ |
|---------| *|-------------- + --------------------------------|
\ sin(2*x)/ | 2 2 |
\ 1 + x 2*sin (x) /
$$\left(\frac{2 \sin^{2}{\left(x \right)}}{\sin{\left(2 x \right)}}\right)^{\operatorname{atan}{\left(x \right)}} \left(\frac{\left(\frac{4 \sin^{4}{\left(x \right)}}{\sin^{2}{\left(2 x \right)}} + 1\right) \sin{\left(2 x \right)} \operatorname{atan}{\left(x \right)}}{2 \sin^{2}{\left(x \right)}} + \frac{\log{\left(\frac{2 \sin^{2}{\left(x \right)}}{\sin{\left(2 x \right)}} \right)}}{x^{2} + 1}\right)$$
/ / 1 \ \
atan(x) |log|------| / 2 \ |
/ 1 \ | \cot(x)/ \1 + cot (x)/*atan(x)|
|------| *|----------- + ---------------------|
\cot(x)/ | 2 cot(x) |
\ 1 + x /
$$\left(\frac{\left(\cot^{2}{\left(x \right)} + 1\right) \operatorname{atan}{\left(x \right)}}{\cot{\left(x \right)}} + \frac{\log{\left(\frac{1}{\cot{\left(x \right)}} \right)}}{x^{2} + 1}\right) \left(\frac{1}{\cot{\left(x \right)}}\right)^{\operatorname{atan}{\left(x \right)}}$$
atan(x) /log(tan(x)) 2*atan(x)\
tan (x)*|----------- + ---------|
| 2 sin(2*x)|
\ 1 + x /
$$\left(\frac{2 \operatorname{atan}{\left(x \right)}}{\sin{\left(2 x \right)}} + \frac{\log{\left(\tan{\left(x \right)} \right)}}{x^{2} + 1}\right) \tan^{\operatorname{atan}{\left(x \right)}}{\left(x \right)}$$
/ / 1 \ \
atan(x) |log|------| |
/ 1 \ | \cot(x)/ / 1 \ |
|------| *|----------- + |1 + -------|*atan(x)*cot(x)|
\cot(x)/ | 2 | 2 | |
\ 1 + x \ cot (x)/ /
$$\left(\left(1 + \frac{1}{\cot^{2}{\left(x \right)}}\right) \cot{\left(x \right)} \operatorname{atan}{\left(x \right)} + \frac{\log{\left(\frac{1}{\cot{\left(x \right)}} \right)}}{x^{2} + 1}\right) \left(\frac{1}{\cot{\left(x \right)}}\right)^{\operatorname{atan}{\left(x \right)}}$$
/ / 2 \ \
| / sec(x) \ | sec (x) | / pi\|
|log|-----------| |1 + ------------|*atan(x)*sec|x - --||
| | / pi\| | 2/ pi\| \ 2 /|
atan(x) | |sec|x - --|| | sec |x - --|| |
/ sec(x) \ | \ \ 2 // \ \ 2 // |
|-----------| *|---------------- + --------------------------------------|
| / pi\| | 2 sec(x) |
|sec|x - --|| \ 1 + x /
\ \ 2 //
$$\left(\frac{\sec{\left(x \right)}}{\sec{\left(x - \frac{\pi}{2} \right)}}\right)^{\operatorname{atan}{\left(x \right)}} \left(\frac{\left(\frac{\sec^{2}{\left(x \right)}}{\sec^{2}{\left(x - \frac{\pi}{2} \right)}} + 1\right) \operatorname{atan}{\left(x \right)} \sec{\left(x - \frac{\pi}{2} \right)}}{\sec{\left(x \right)}} + \frac{\log{\left(\frac{\sec{\left(x \right)}}{\sec{\left(x - \frac{\pi}{2} \right)}} \right)}}{x^{2} + 1}\right)$$
/ / /pi \\ \
atan(x) | |csc|-- - x|| |
/ /pi \\ | | \2 /| |
|csc|-- - x|| |log|-----------| |
| \2 /| | \ csc(x) / |
|-----------| *|---------------- + 2*atan(x)*csc(2*x)|
\ csc(x) / | 2 |
\ 1 + x /
$$\left(\frac{\csc{\left(- x + \frac{\pi}{2} \right)}}{\csc{\left(x \right)}}\right)^{\operatorname{atan}{\left(x \right)}} \left(2 \operatorname{atan}{\left(x \right)} \csc{\left(2 x \right)} + \frac{\log{\left(\frac{\csc{\left(- x + \frac{\pi}{2} \right)}}{\csc{\left(x \right)}} \right)}}{x^{2} + 1}\right)$$
/ / 2 \ \
| | sin (x)| |
| /sin(x)\ |1 + -------|*atan(x)*cos(x)|
atan(x) |log|------| | 2 | |
/sin(x)\ | \cos(x)/ \ cos (x)/ |
|------| *|----------- + ----------------------------|
\cos(x)/ | 2 sin(x) |
\ 1 + x /
$$\left(\frac{\sin{\left(x \right)}}{\cos{\left(x \right)}}\right)^{\operatorname{atan}{\left(x \right)}} \left(\frac{\left(\frac{\sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 1\right) \cos{\left(x \right)} \operatorname{atan}{\left(x \right)}}{\sin{\left(x \right)}} + \frac{\log{\left(\frac{\sin{\left(x \right)}}{\cos{\left(x \right)}} \right)}}{x^{2} + 1}\right)$$
/ / sec(x) \ \
|log|-----------| |
| | / pi\| |
atan(x) | |sec|x - --|| |
/ sec(x) \ | \ \ 2 // / pi\|
|-----------| *|---------------- + 2*atan(x)*sec|2*x - --||
| / pi\| | 2 \ 2 /|
|sec|x - --|| \ 1 + x /
\ \ 2 //
$$\left(\frac{\sec{\left(x \right)}}{\sec{\left(x - \frac{\pi}{2} \right)}}\right)^{\operatorname{atan}{\left(x \right)}} \left(2 \operatorname{atan}{\left(x \right)} \sec{\left(2 x - \frac{\pi}{2} \right)} + \frac{\log{\left(\frac{\sec{\left(x \right)}}{\sec{\left(x - \frac{\pi}{2} \right)}} \right)}}{x^{2} + 1}\right)$$
/ / 2/pi \\ \
| / /pi \\ | csc |-- - x|| |
atan(x) | |csc|-- - x|| | \2 /| |
/ /pi \\ | | \2 /| |1 + ------------|*atan(x)*csc(x)|
|csc|-- - x|| |log|-----------| | 2 | |
| \2 /| | \ csc(x) / \ csc (x) / |
|-----------| *|---------------- + ---------------------------------|
\ csc(x) / | 2 /pi \ |
| 1 + x csc|-- - x| |
\ \2 / /
$$\left(\frac{\csc{\left(- x + \frac{\pi}{2} \right)}}{\csc{\left(x \right)}}\right)^{\operatorname{atan}{\left(x \right)}} \left(\frac{\left(1 + \frac{\csc^{2}{\left(- x + \frac{\pi}{2} \right)}}{\csc^{2}{\left(x \right)}}\right) \operatorname{atan}{\left(x \right)} \csc{\left(x \right)}}{\csc{\left(- x + \frac{\pi}{2} \right)}} + \frac{\log{\left(\frac{\csc{\left(- x + \frac{\pi}{2} \right)}}{\csc{\left(x \right)}} \right)}}{x^{2} + 1}\right)$$
/ / 2 \ \
atan(x) | |2*sin (x)| |
/ 2 \ |log|---------| |
|2*sin (x)| | \ sin(2*x)/ 2*atan(x)|
|---------| *|-------------- + ---------|
\ sin(2*x)/ | 2 sin(2*x)|
\ 1 + x /
$$\left(\frac{2 \sin^{2}{\left(x \right)}}{\sin{\left(2 x \right)}}\right)^{\operatorname{atan}{\left(x \right)}} \left(\frac{2 \operatorname{atan}{\left(x \right)}}{\sin{\left(2 x \right)}} + \frac{\log{\left(\frac{2 \sin^{2}{\left(x \right)}}{\sin{\left(2 x \right)}} \right)}}{x^{2} + 1}\right)$$
/ / 2/ pi\\ \
| / / pi\\ | cos |x - --|| |
atan(x) | |cos|x - --|| | \ 2 /| |
/ / pi\\ | | \ 2 /| |1 + ------------|*atan(x)*cos(x)|
|cos|x - --|| |log|-----------| | 2 | |
| \ 2 /| | \ cos(x) / \ cos (x) / |
|-----------| *|---------------- + ---------------------------------|
\ cos(x) / | 2 / pi\ |
| 1 + x cos|x - --| |
\ \ 2 / /
$$\left(\frac{\cos{\left(x - \frac{\pi}{2} \right)}}{\cos{\left(x \right)}}\right)^{\operatorname{atan}{\left(x \right)}} \left(\frac{\left(1 + \frac{\cos^{2}{\left(x - \frac{\pi}{2} \right)}}{\cos^{2}{\left(x \right)}}\right) \cos{\left(x \right)} \operatorname{atan}{\left(x \right)}}{\cos{\left(x - \frac{\pi}{2} \right)}} + \frac{\log{\left(\frac{\cos{\left(x - \frac{\pi}{2} \right)}}{\cos{\left(x \right)}} \right)}}{x^{2} + 1}\right)$$
(cos(x - pi/2)/cos(x))^atan(x)*(log(cos(x - pi/2)/cos(x))/(1 + x^2) + (1 + cos(x - pi/2)^2/cos(x)^2)*atan(x)*cos(x)/cos(x - pi/2))
Denominador racional
[src]
atan(x) / / 2\ / 2 \ \
tan (x)*\log(tan(x))*tan(x) + \1 + x /*\1 + tan (x)/*atan(x)/
-------------------------------------------------------------------
/ 2\
\1 + x /*tan(x)
$$\frac{\left(\left(x^{2} + 1\right) \left(\tan^{2}{\left(x \right)} + 1\right) \operatorname{atan}{\left(x \right)} + \log{\left(\tan{\left(x \right)} \right)} \tan{\left(x \right)}\right) \tan^{\operatorname{atan}{\left(x \right)}}{\left(x \right)}}{\left(x^{2} + 1\right) \tan{\left(x \right)}}$$
tan(x)^atan(x)*(log(tan(x))*tan(x) + (1 + x^2)*(1 + tan(x)^2)*atan(x))/((1 + x^2)*tan(x))
Denominador común
[src]
atan(x) 2 atan(x) atan(x)
tan (x)*atan(x) + x *tan (x)*atan(x) + tan (x)*log(tan(x))*tan(x) atan(x)
----------------------------------------------------------------------------------- + tan (x)*atan(x)*tan(x)
2
x *tan(x) + tan(x)
$$\tan{\left(x \right)} \tan^{\operatorname{atan}{\left(x \right)}}{\left(x \right)} \operatorname{atan}{\left(x \right)} + \frac{x^{2} \tan^{\operatorname{atan}{\left(x \right)}}{\left(x \right)} \operatorname{atan}{\left(x \right)} + \log{\left(\tan{\left(x \right)} \right)} \tan{\left(x \right)} \tan^{\operatorname{atan}{\left(x \right)}}{\left(x \right)} + \tan^{\operatorname{atan}{\left(x \right)}}{\left(x \right)} \operatorname{atan}{\left(x \right)}}{x^{2} \tan{\left(x \right)} + \tan{\left(x \right)}}$$
(tan(x)^atan(x)*atan(x) + x^2*tan(x)^atan(x)*atan(x) + tan(x)^atan(x)*log(tan(x))*tan(x))/(x^2*tan(x) + tan(x)) + tan(x)^atan(x)*atan(x)*tan(x)