Sr Examen
Simplificación de expresiones/
Descomposición en fracciones simples/
cos(atan(x))/sqrt(1-x^2)-x*asin(x)/(1+x^2)^(3/2)
¿Cómo vas a descomponer esta cos(atan(x))/sqrt(1-x^2)-x*asin(x)/(1+x^2)^(3/2) expresión en fracciones?
Solución
Ha introducido
[src]
cos(atan(x)) x*asin(x) ------------ - ----------- ________ 3/2 / 2 / 2\ \/ 1 - x \1 + x /
$$- \frac{x \operatorname{asin}{\left(x \right)}}{\left(x^{2} + 1\right)^{\frac{3}{2}}} + \frac{\cos{\left(\operatorname{atan}{\left(x \right)} \right)}}{\sqrt{1 - x^{2}}}$$
cos(atan(x))/sqrt(1 - x^2) - x*asin(x)/(1 + x^2)^(3/2)
Simplificación general
[src]
________
2 / 2
1 + x - x*\/ 1 - x *asin(x)
------------------------------
3/2 ________
/ 2\ / 2
\1 + x / *\/ 1 - x
$$\frac{x^{2} - x \sqrt{1 - x^{2}} \operatorname{asin}{\left(x \right)} + 1}{\sqrt{1 - x^{2}} \left(x^{2} + 1\right)^{\frac{3}{2}}}$$
(1 + x^2 - x*sqrt(1 - x^2)*asin(x))/((1 + x^2)^(3/2)*sqrt(1 - x^2))
Respuesta numérica
[src]
(1.0 - x^2)^(-0.5)*cos(atan(x)) - x*(1.0 + x^2)^(-1.5)*asin(x)
(1.0 - x^2)^(-0.5)*cos(atan(x)) - x*(1.0 + x^2)^(-1.5)*asin(x)
Unión de expresiones racionales
[src]
________
2 / 2
1 + x - x*\/ 1 - x *asin(x)
------------------------------
3/2 ________
/ 2\ / 2
\1 + x / *\/ 1 - x
$$\frac{x^{2} - x \sqrt{1 - x^{2}} \operatorname{asin}{\left(x \right)} + 1}{\sqrt{1 - x^{2}} \left(x^{2} + 1\right)^{\frac{3}{2}}}$$
(1 + x^2 - x*sqrt(1 - x^2)*asin(x))/((1 + x^2)^(3/2)*sqrt(1 - x^2))
Potencias
[src]
1 x*asin(x) ---------------------- - ----------- ___________________ 3/2 / / 2\ / 2\ / 2\ \/ \1 + x /*\1 - x / \1 + x /
$$- \frac{x \operatorname{asin}{\left(x \right)}}{\left(x^{2} + 1\right)^{\frac{3}{2}}} + \frac{1}{\sqrt{\left(1 - x^{2}\right) \left(x^{2} + 1\right)}}$$
I*atan(x) -I*atan(x)
e e
---------- + -----------
2 2 x*asin(x)
------------------------ - -----------
________ 3/2
/ 2 / 2\
\/ 1 - x \1 + x /
$$- \frac{x \operatorname{asin}{\left(x \right)}}{\left(x^{2} + 1\right)^{\frac{3}{2}}} + \frac{\frac{e^{i \operatorname{atan}{\left(x \right)}}}{2} + \frac{e^{- i \operatorname{atan}{\left(x \right)}}}{2}}{\sqrt{1 - x^{2}}}$$
1 x*asin(x) ----------------------- - ----------- ________ ________ 3/2 / 2 / 2 / 2\ \/ 1 + x *\/ 1 - x \1 + x /
$$- \frac{x \operatorname{asin}{\left(x \right)}}{\left(x^{2} + 1\right)^{\frac{3}{2}}} + \frac{1}{\sqrt{1 - x^{2}} \sqrt{x^{2} + 1}}$$
1/(sqrt(1 + x^2)*sqrt(1 - x^2)) - x*asin(x)/(1 + x^2)^(3/2)
Combinatoria
[src]
________
2 / 2
1 + x - x*\/ 1 - x *asin(x)
---------------------------------
3/2
___________________ / 2\
\/ -(1 + x)*(-1 + x) *\1 + x /
$$\frac{x^{2} - x \sqrt{1 - x^{2}} \operatorname{asin}{\left(x \right)} + 1}{\sqrt{- \left(x - 1\right) \left(x + 1\right)} \left(x^{2} + 1\right)^{\frac{3}{2}}}$$
(1 + x^2 - x*sqrt(1 - x^2)*asin(x))/(sqrt(-(1 + x)*(-1 + x))*(1 + x^2)^(3/2))
Denominador común
[src]
________ ________ ________
/ 2 2 / 2 / 2
\/ 1 + x *cos(atan(x)) + x *\/ 1 + x *cos(atan(x)) - x*\/ 1 - x *asin(x)
------------------------------------------------------------------------------
________ ________ ________ ________
/ 2 / 2 2 / 2 / 2
\/ 1 + x *\/ 1 - x + x *\/ 1 + x *\/ 1 - x
$$\frac{x^{2} \sqrt{x^{2} + 1} \cos{\left(\operatorname{atan}{\left(x \right)} \right)} - x \sqrt{1 - x^{2}} \operatorname{asin}{\left(x \right)} + \sqrt{x^{2} + 1} \cos{\left(\operatorname{atan}{\left(x \right)} \right)}}{x^{2} \sqrt{1 - x^{2}} \sqrt{x^{2} + 1} + \sqrt{1 - x^{2}} \sqrt{x^{2} + 1}}$$
(sqrt(1 + x^2)*cos(atan(x)) + x^2*sqrt(1 + x^2)*cos(atan(x)) - x*sqrt(1 - x^2)*asin(x))/(sqrt(1 + x^2)*sqrt(1 - x^2) + x^2*sqrt(1 + x^2)*sqrt(1 - x^2))
Denominador racional
[src]
________ ________ ________ ________ ________ ________
/ 2 / 2 / 2 2 / 2 / 2 3 / 2
- \/ 1 + x *\/ 1 - x + x*\/ 1 + x *asin(x) - x *\/ 1 + x *\/ 1 - x - x *\/ 1 + x *asin(x)
-------------------------------------------------------------------------------------------------------
/ 2\ / 4 2\
\-1 + x /*\1 + x + 2*x /
$$\frac{- x^{3} \sqrt{x^{2} + 1} \operatorname{asin}{\left(x \right)} - x^{2} \sqrt{1 - x^{2}} \sqrt{x^{2} + 1} + x \sqrt{x^{2} + 1} \operatorname{asin}{\left(x \right)} - \sqrt{1 - x^{2}} \sqrt{x^{2} + 1}}{\left(x^{2} - 1\right) \left(x^{4} + 2 x^{2} + 1\right)}$$
(-sqrt(1 + x^2)*sqrt(1 - x^2) + x*sqrt(1 + x^2)*asin(x) - x^2*sqrt(1 + x^2)*sqrt(1 - x^2) - x^3*sqrt(1 + x^2)*asin(x))/((-1 + x^2)*(1 + x^4 + 2*x^2))
Parte trigonométrica
[src]
/ 2/atan(x) pi\\
| cos |------- - --||
/ 1 \ | \ 2 2 /|
|1 + -----------|*|1 - ------------------|
| ________| | 2/atan(x)\ |
| / 2 | | cos |-------| |
\ \/ 1 + x / \ \ 2 / / x*asin(x)
------------------------------------------ - -----------
________ 3/2
/ 2 / 2\
2*\/ 1 - x \1 + x /
$$- \frac{x \operatorname{asin}{\left(x \right)}}{\left(x^{2} + 1\right)^{\frac{3}{2}}} + \frac{\left(1 + \frac{1}{\sqrt{x^{2} + 1}}\right) \left(- \frac{\cos^{2}{\left(\frac{\operatorname{atan}{\left(x \right)}}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{\operatorname{atan}{\left(x \right)}}{2} \right)}} + 1\right)}{2 \sqrt{1 - x^{2}}}$$
/ 4/atan(x)\ / 2\\
| 4*sin |-------|*\1 + x /|
/ 1 \ | \ 2 / |
|1 + -----------|*|1 - ------------------------|
| ________| | 2 |
| / 2 | \ x /
\ \/ 1 + x / x*asin(x)
------------------------------------------------ - -----------
________ 3/2
/ 2 / 2\
2*\/ 1 - x \1 + x /
$$- \frac{x \operatorname{asin}{\left(x \right)}}{\left(x^{2} + 1\right)^{\frac{3}{2}}} + \frac{\left(1 - \frac{4 \left(x^{2} + 1\right) \sin^{4}{\left(\frac{\operatorname{atan}{\left(x \right)}}{2} \right)}}{x^{2}}\right) \left(1 + \frac{1}{\sqrt{x^{2} + 1}}\right)}{2 \sqrt{1 - x^{2}}}$$
/ 2/atan(x)\ \
| cos |-------| |
/ 1 \ | \ 2 / |
|1 - -----------|*|-1 + ------------------|
| ________| | 2/atan(x) pi\|
| / 2 | | cos |------- - --||
\ \/ 1 + x / \ \ 2 2 // x*asin(x)
------------------------------------------- - -----------
________ 3/2
/ 2 / 2\
2*\/ 1 - x \1 + x /
$$- \frac{x \operatorname{asin}{\left(x \right)}}{\left(x^{2} + 1\right)^{\frac{3}{2}}} + \frac{\left(-1 + \frac{\cos^{2}{\left(\frac{\operatorname{atan}{\left(x \right)}}{2} \right)}}{\cos^{2}{\left(\frac{\operatorname{atan}{\left(x \right)}}{2} - \frac{\pi}{2} \right)}}\right) \left(1 - \frac{1}{\sqrt{x^{2} + 1}}\right)}{2 \sqrt{1 - x^{2}}}$$
/ 1 \ / 1 \
|1 + -----------|*|1 - -------------|
| ________| | 2/atan(x)\|
| / 2 | | cot |-------||
\ \/ 1 + x / \ \ 2 // x*asin(x)
------------------------------------- - -----------
________ 3/2
/ 2 / 2\
2*\/ 1 - x \1 + x /
$$- \frac{x \operatorname{asin}{\left(x \right)}}{\left(x^{2} + 1\right)^{\frac{3}{2}}} + \frac{\left(1 + \frac{1}{\sqrt{x^{2} + 1}}\right) \left(1 - \frac{1}{\cot^{2}{\left(\frac{\operatorname{atan}{\left(x \right)}}{2} \right)}}\right)}{2 \sqrt{1 - x^{2}}}$$
2/atan(x)\
1 - tan |-------|
\ 2 / x*asin(x)
------------------------------- - -----------
________ 3/2
/ 2/atan(x)\\ / 2 / 2\
|1 + tan |-------||*\/ 1 - x \1 + x /
\ \ 2 //
$$- \frac{x \operatorname{asin}{\left(x \right)}}{\left(x^{2} + 1\right)^{\frac{3}{2}}} + \frac{1 - \tan^{2}{\left(\frac{\operatorname{atan}{\left(x \right)}}{2} \right)}}{\sqrt{1 - x^{2}} \left(\tan^{2}{\left(\frac{\operatorname{atan}{\left(x \right)}}{2} \right)} + 1\right)}$$
/ 1 \ / 2/atan(x)\\
|1 - -----------|*|-1 + cot |-------||
| ________| \ \ 2 //
| / 2 |
\ \/ 1 + x / x*asin(x)
-------------------------------------- - -----------
________ 3/2
/ 2 / 2\
2*\/ 1 - x \1 + x /
$$- \frac{x \operatorname{asin}{\left(x \right)}}{\left(x^{2} + 1\right)^{\frac{3}{2}}} + \frac{\left(1 - \frac{1}{\sqrt{x^{2} + 1}}\right) \left(\cot^{2}{\left(\frac{\operatorname{atan}{\left(x \right)}}{2} \right)} - 1\right)}{2 \sqrt{1 - x^{2}}}$$
/ 1 \ / 1 \
|1 - -----------|*|-1 + -------------|
| ________| | 2/atan(x)\|
| / 2 | | tan |-------||
\ \/ 1 + x / \ \ 2 // x*asin(x)
-------------------------------------- - -----------
________ 3/2
/ 2 / 2\
2*\/ 1 - x \1 + x /
$$- \frac{x \operatorname{asin}{\left(x \right)}}{\left(x^{2} + 1\right)^{\frac{3}{2}}} + \frac{\left(-1 + \frac{1}{\tan^{2}{\left(\frac{\operatorname{atan}{\left(x \right)}}{2} \right)}}\right) \left(1 - \frac{1}{\sqrt{x^{2} + 1}}\right)}{2 \sqrt{1 - x^{2}}}$$
/ ________ \
| / 2 / 1 \|
| \/ 1 + x *|1 - -----------||
| | ________||
| | / 2 ||
/ 1 \ | \ \/ 1 + x /|
|1 + -----------|*|1 - -----------------------------|
| ________| | ________ |
| / 2 | | / 2 |
\ \/ 1 + x / \ 1 + \/ 1 + x / x*asin(x)
----------------------------------------------------- - -----------
________ 3/2
/ 2 / 2\
2*\/ 1 - x \1 + x /
$$- \frac{x \operatorname{asin}{\left(x \right)}}{\left(x^{2} + 1\right)^{\frac{3}{2}}} + \frac{\left(1 + \frac{1}{\sqrt{x^{2} + 1}}\right) \left(- \frac{\left(1 - \frac{1}{\sqrt{x^{2} + 1}}\right) \sqrt{x^{2} + 1}}{\sqrt{x^{2} + 1} + 1} + 1\right)}{2 \sqrt{1 - x^{2}}}$$
/ 2/atan(x)\ \
| sec |-------| |
/ 1 \ | \ 2 / |
|1 + -----------|*|1 - ------------------|
| ________| | 2/atan(x) pi\|
| / 2 | | sec |------- - --||
\ \/ 1 + x / \ \ 2 2 // x*asin(x)
------------------------------------------ - -----------
________ 3/2
/ 2 / 2\
2*\/ 1 - x \1 + x /
$$- \frac{x \operatorname{asin}{\left(x \right)}}{\left(x^{2} + 1\right)^{\frac{3}{2}}} + \frac{\left(1 - \frac{\sec^{2}{\left(\frac{\operatorname{atan}{\left(x \right)}}{2} \right)}}{\sec^{2}{\left(\frac{\operatorname{atan}{\left(x \right)}}{2} - \frac{\pi}{2} \right)}}\right) \left(1 + \frac{1}{\sqrt{x^{2} + 1}}\right)}{2 \sqrt{1 - x^{2}}}$$
/ 2/atan(x)\ \
| csc |-------| |
/ 1 \ | \ 2 / |
|1 - -----------|*|-1 + ------------------|
| ________| | 2/pi atan(x)\|
| / 2 | | csc |-- - -------||
\ \/ 1 + x / \ \2 2 // x*asin(x)
------------------------------------------- - -----------
________ 3/2
/ 2 / 2\
2*\/ 1 - x \1 + x /
$$- \frac{x \operatorname{asin}{\left(x \right)}}{\left(x^{2} + 1\right)^{\frac{3}{2}}} + \frac{\left(-1 + \frac{\csc^{2}{\left(\frac{\operatorname{atan}{\left(x \right)}}{2} \right)}}{\csc^{2}{\left(- \frac{\operatorname{atan}{\left(x \right)}}{2} + \frac{\pi}{2} \right)}}\right) \left(1 - \frac{1}{\sqrt{x^{2} + 1}}\right)}{2 \sqrt{1 - x^{2}}}$$
1 x*asin(x) ----------------------- - ----------- ________ ________ 3/2 / 2 / 2 / 2\ \/ 1 + x *\/ 1 - x \1 + x /
$$- \frac{x \operatorname{asin}{\left(x \right)}}{\left(x^{2} + 1\right)^{\frac{3}{2}}} + \frac{1}{\sqrt{1 - x^{2}} \sqrt{x^{2} + 1}}$$
2/atan(x)\
-1 + cot |-------|
\ 2 / x*asin(x)
------------------------------- - -----------
________ 3/2
/ 2/atan(x)\\ / 2 / 2\
|1 + cot |-------||*\/ 1 - x \1 + x /
\ \ 2 //
$$- \frac{x \operatorname{asin}{\left(x \right)}}{\left(x^{2} + 1\right)^{\frac{3}{2}}} + \frac{\cot^{2}{\left(\frac{\operatorname{atan}{\left(x \right)}}{2} \right)} - 1}{\sqrt{1 - x^{2}} \left(\cot^{2}{\left(\frac{\operatorname{atan}{\left(x \right)}}{2} \right)} + 1\right)}$$
/ 2 \
/ 1 \ | x |
|1 - -----------|*|-1 + -------------------|
| ________| | 2|
| / 2 | | / ________\ |
\ \/ 1 + x / | | / 2 | |
\ \-1 + \/ 1 + x / / x*asin(x)
-------------------------------------------- - -----------
________ 3/2
/ 2 / 2\
2*\/ 1 - x \1 + x /
$$- \frac{x \operatorname{asin}{\left(x \right)}}{\left(x^{2} + 1\right)^{\frac{3}{2}}} + \frac{\left(1 - \frac{1}{\sqrt{x^{2} + 1}}\right) \left(\frac{x^{2}}{\left(\sqrt{x^{2} + 1} - 1\right)^{2}} - 1\right)}{2 \sqrt{1 - x^{2}}}$$
/ ________ \
| / 2 / 1 \|
| \/ 1 + x *|1 + -----------||
| | ________||
| | / 2 ||
/ 1 \ | \ \/ 1 + x /|
|1 - -----------|*|-1 + -----------------------------|
| ________| | ________ |
| / 2 | | / 2 |
\ \/ 1 + x / \ -1 + \/ 1 + x / x*asin(x)
------------------------------------------------------ - -----------
________ 3/2
/ 2 / 2\
2*\/ 1 - x \1 + x /
$$- \frac{x \operatorname{asin}{\left(x \right)}}{\left(x^{2} + 1\right)^{\frac{3}{2}}} + \frac{\left(1 - \frac{1}{\sqrt{x^{2} + 1}}\right) \left(\frac{\left(1 + \frac{1}{\sqrt{x^{2} + 1}}\right) \sqrt{x^{2} + 1}}{\sqrt{x^{2} + 1} - 1} - 1\right)}{2 \sqrt{1 - x^{2}}}$$
/ 2/pi atan(x)\\
| csc |-- - -------||
/ 1 \ | \2 2 /|
|1 + -----------|*|1 - ------------------|
| ________| | 2/atan(x)\ |
| / 2 | | csc |-------| |
\ \/ 1 + x / \ \ 2 / / x*asin(x)
------------------------------------------ - -----------
________ 3/2
/ 2 / 2\
2*\/ 1 - x \1 + x /
$$- \frac{x \operatorname{asin}{\left(x \right)}}{\left(x^{2} + 1\right)^{\frac{3}{2}}} + \frac{\left(1 + \frac{1}{\sqrt{x^{2} + 1}}\right) \left(- \frac{\csc^{2}{\left(- \frac{\operatorname{atan}{\left(x \right)}}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{\operatorname{atan}{\left(x \right)}}{2} \right)}} + 1\right)}{2 \sqrt{1 - x^{2}}}$$
/ 2 \
/ 1 \ | x |
|1 - -----------|*|-1 + ------------------------|
| ________| | / 2\ 4/atan(x)\|
| / 2 | | 4*\1 + x /*sin |-------||
\ \/ 1 + x / \ \ 2 // x*asin(x)
------------------------------------------------- - -----------
________ 3/2
/ 2 / 2\
2*\/ 1 - x \1 + x /
$$- \frac{x \operatorname{asin}{\left(x \right)}}{\left(x^{2} + 1\right)^{\frac{3}{2}}} + \frac{\left(1 - \frac{1}{\sqrt{x^{2} + 1}}\right) \left(\frac{x^{2}}{4 \left(x^{2} + 1\right) \sin^{4}{\left(\frac{\operatorname{atan}{\left(x \right)}}{2} \right)}} - 1\right)}{2 \sqrt{1 - x^{2}}}$$
/ 2/atan(x) pi\\
| sec |------- - --||
/ 1 \ | \ 2 2 /|
|1 - -----------|*|-1 + ------------------|
| ________| | 2/atan(x)\ |
| / 2 | | sec |-------| |
\ \/ 1 + x / \ \ 2 / / x*asin(x)
------------------------------------------- - -----------
________ 3/2
/ 2 / 2\
2*\/ 1 - x \1 + x /
$$- \frac{x \operatorname{asin}{\left(x \right)}}{\left(x^{2} + 1\right)^{\frac{3}{2}}} + \frac{\left(1 - \frac{1}{\sqrt{x^{2} + 1}}\right) \left(\frac{\sec^{2}{\left(\frac{\operatorname{atan}{\left(x \right)}}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{\operatorname{atan}{\left(x \right)}}{2} \right)}} - 1\right)}{2 \sqrt{1 - x^{2}}}$$
/ 2\
| / ________\ |
| | / 2 | |
/ 1 \ | \-1 + \/ 1 + x / |
|1 + -----------|*|1 - -------------------|
| ________| | 2 |
| / 2 | \ x /
\ \/ 1 + x / x*asin(x)
------------------------------------------- - -----------
________ 3/2
/ 2 / 2\
2*\/ 1 - x \1 + x /
$$- \frac{x \operatorname{asin}{\left(x \right)}}{\left(x^{2} + 1\right)^{\frac{3}{2}}} + \frac{\left(1 - \frac{\left(\sqrt{x^{2} + 1} - 1\right)^{2}}{x^{2}}\right) \left(1 + \frac{1}{\sqrt{x^{2} + 1}}\right)}{2 \sqrt{1 - x^{2}}}$$
/ 1 \ / 2/atan(x)\\
|1 + -----------|*|1 - tan |-------||
| ________| \ \ 2 //
| / 2 |
\ \/ 1 + x / x*asin(x)
------------------------------------- - -----------
________ 3/2
/ 2 / 2\
2*\/ 1 - x \1 + x /
$$- \frac{x \operatorname{asin}{\left(x \right)}}{\left(x^{2} + 1\right)^{\frac{3}{2}}} + \frac{\left(1 + \frac{1}{\sqrt{x^{2} + 1}}\right) \left(1 - \tan^{2}{\left(\frac{\operatorname{atan}{\left(x \right)}}{2} \right)}\right)}{2 \sqrt{1 - x^{2}}}$$
/pi \
sin|-- + atan(x)|
\2 / x*asin(x)
----------------- - -----------
________ 3/2
/ 2 / 2\
\/ 1 - x \1 + x /
$$- \frac{x \operatorname{asin}{\left(x \right)}}{\left(x^{2} + 1\right)^{\frac{3}{2}}} + \frac{\sin{\left(\operatorname{atan}{\left(x \right)} + \frac{\pi}{2} \right)}}{\sqrt{1 - x^{2}}}$$
sin(pi/2 + atan(x))/sqrt(1 - x^2) - x*asin(x)/(1 + x^2)^(3/2)