Sr Examen
Simplificación de expresiones/
Descomposición en fracciones simples/
cos(x)^(x^2/(x+1))*((-x^2/(x+1)^2+2*x/(x+1))*log(cos(x))-x^2*sin(x)/((x+1)*cos(x)))
¿Cómo vas a descomponer esta cos(x)^(x^2/(x+1))*((-x^2/(x+1)^2+2*x/(x+1))*log(cos(x))-x^2*sin(x)/((x+1)*cos(x))) expresión en fracciones?
Solución
Ha introducido
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2
x
----- // 2 \ 2 \
x + 1 || -x 2*x | x *sin(x) |
(cos(x)) *||-------- + -----|*log(cos(x)) - --------------|
|| 2 x + 1| (x + 1)*cos(x)|
\\(x + 1) / /
$$\left(- \frac{x^{2} \sin{\left(x \right)}}{\left(x + 1\right) \cos{\left(x \right)}} + \left(\frac{2 x}{x + 1} + \frac{\left(-1\right) x^{2}}{\left(x + 1\right)^{2}}\right) \log{\left(\cos{\left(x \right)} \right)}\right) \cos^{\frac{x^{2}}{x + 1}}{\left(x \right)}$$
cos(x)^(x^2/(x + 1))*(((-x^2)/(x + 1)^2 + (2*x)/(x + 1))*log(cos(x)) - x^2*sin(x)/((x + 1)*cos(x)))
Simplificación general
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2
-1 + x - x
-----------
1 + x
x*(cos(x)) *((2 + x)*cos(x)*log(cos(x)) - x*(1 + x)*sin(x))
---------------------------------------------------------------------
2
(1 + x)
$$\frac{x \left(- x \left(x + 1\right) \sin{\left(x \right)} + \left(x + 2\right) \log{\left(\cos{\left(x \right)} \right)} \cos{\left(x \right)}\right) \cos^{\frac{x^{2} - x - 1}{x + 1}}{\left(x \right)}}{\left(x + 1\right)^{2}}$$
x*cos(x)^((-1 + x^2 - x)/(1 + x))*((2 + x)*cos(x)*log(cos(x)) - x*(1 + x)*sin(x))/(1 + x)^2
Combinatoria
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2
x
-----
1 + x / 2 \
-x*(cos(x)) *\x*sin(x) + x *sin(x) - 2*cos(x)*log(cos(x)) - x*cos(x)*log(cos(x))/
--------------------------------------------------------------------------------------
2
(1 + x) *cos(x)
$$- \frac{x \left(x^{2} \sin{\left(x \right)} - x \log{\left(\cos{\left(x \right)} \right)} \cos{\left(x \right)} + x \sin{\left(x \right)} - 2 \log{\left(\cos{\left(x \right)} \right)} \cos{\left(x \right)}\right) \cos^{\frac{x^{2}}{x + 1}}{\left(x \right)}}{\left(x + 1\right)^{2} \cos{\left(x \right)}}$$
-x*cos(x)^(x^2/(1 + x))*(x*sin(x) + x^2*sin(x) - 2*cos(x)*log(cos(x)) - x*cos(x)*log(cos(x)))/((1 + x)^2*cos(x))
Respuesta numérica
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cos(x)^(x^2/(1.0 + x))*((-x^2/(1.0 + x)^2 + 2.0*x/(1.0 + x))*log(cos(x)) - x^2*sin(x)/((1.0 + x)*cos(x)))
cos(x)^(x^2/(1.0 + x))*((-x^2/(1.0 + x)^2 + 2.0*x/(1.0 + x))*log(cos(x)) - x^2*sin(x)/((1.0 + x)*cos(x)))
Denominador racional
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2
x
-----
x + 1 / 2 3 / 2 2\ \
(cos(x)) *\- x *(1 + x) *sin(x) + (1 + x)*\- x *(1 + x) + 2*x*(1 + x) /*cos(x)*log(cos(x))/
-----------------------------------------------------------------------------------------------
4
(1 + x) *cos(x)
$$\frac{\left(- x^{2} \left(x + 1\right)^{3} \sin{\left(x \right)} + \left(x + 1\right) \left(- x^{2} \left(x + 1\right) + 2 x \left(x + 1\right)^{2}\right) \log{\left(\cos{\left(x \right)} \right)} \cos{\left(x \right)}\right) \cos^{\frac{x^{2}}{x + 1}}{\left(x \right)}}{\left(x + 1\right)^{4} \cos{\left(x \right)}}$$
cos(x)^(x^2/(x + 1))*(-x^2*(1 + x)^3*sin(x) + (1 + x)*(-x^2*(1 + x) + 2*x*(1 + x)^2)*cos(x)*log(cos(x)))/((1 + x)^4*cos(x))
Denominador común
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2 2 2
2 x x x
x ----- ----- -----
----- 2 1 + x 3 1 + x 1 + x
1 + x x *(cos(x)) *sin(x) + x *(cos(x)) *sin(x) + (cos(x)) *cos(x)*log(cos(x))
(cos(x)) *log(cos(x)) - ------------------------------------------------------------------------------------
2
x *cos(x) + 2*x*cos(x) + cos(x)
$$\log{\left(\cos{\left(x \right)} \right)} \cos^{\frac{x^{2}}{x + 1}}{\left(x \right)} - \frac{x^{3} \sin{\left(x \right)} \cos^{\frac{x^{2}}{x + 1}}{\left(x \right)} + x^{2} \sin{\left(x \right)} \cos^{\frac{x^{2}}{x + 1}}{\left(x \right)} + \log{\left(\cos{\left(x \right)} \right)} \cos{\left(x \right)} \cos^{\frac{x^{2}}{x + 1}}{\left(x \right)}}{x^{2} \cos{\left(x \right)} + 2 x \cos{\left(x \right)} + \cos{\left(x \right)}}$$
cos(x)^(x^2/(1 + x))*log(cos(x)) - (x^2*cos(x)^(x^2/(1 + x))*sin(x) + x^3*cos(x)^(x^2/(1 + x))*sin(x) + cos(x)^(x^2/(1 + x))*cos(x)*log(cos(x)))/(x^2*cos(x) + 2*x*cos(x) + cos(x))
Potencias
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2
x
-----
1 + x
/ I*x -I*x\ // 2 \ / I*x -I*x\ 2 / -I*x I*x\ \
|e e | || x 2*x | |e e | I*x *\- e + e / |
|---- + -----| *||- -------- + -----|*log|---- + -----| + ------------------------|
\ 2 2 / || 2 1 + x| \ 2 2 / / I*x -I*x\|
|\ (1 + x) / |e e ||
| 2*(1 + x)*|---- + -----||
\ \ 2 2 //
$$\left(\frac{i x^{2} \left(e^{i x} - e^{- i x}\right)}{2 \left(x + 1\right) \left(\frac{e^{i x}}{2} + \frac{e^{- i x}}{2}\right)} + \left(- \frac{x^{2}}{\left(x + 1\right)^{2}} + \frac{2 x}{x + 1}\right) \log{\left(\frac{e^{i x}}{2} + \frac{e^{- i x}}{2} \right)}\right) \left(\frac{e^{i x}}{2} + \frac{e^{- i x}}{2}\right)^{\frac{x^{2}}{x + 1}}$$
2
x
----- // 2 \ 2 \
1 + x || x 2*x | x *sin(x) |
(cos(x)) *||- -------- + -----|*log(cos(x)) - --------------|
|| 2 1 + x| (1 + x)*cos(x)|
\\ (1 + x) / /
$$\left(- \frac{x^{2} \sin{\left(x \right)}}{\left(x + 1\right) \cos{\left(x \right)}} + \left(- \frac{x^{2}}{\left(x + 1\right)^{2}} + \frac{2 x}{x + 1}\right) \log{\left(\cos{\left(x \right)} \right)}\right) \cos^{\frac{x^{2}}{x + 1}}{\left(x \right)}$$
cos(x)^(x^2/(1 + x))*((-x^2/(1 + x)^2 + 2*x/(1 + x))*log(cos(x)) - x^2*sin(x)/((1 + x)*cos(x)))
Parte trigonométrica
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2
x
-----
1 + x // 2 \ 2 \
/ / pi\\ || x 2*x | / / pi\\ x *sin(x) |
|sin|x + --|| *||- -------- + -----|*log|sin|x + --|| - -------------------|
\ \ 2 // || 2 1 + x| \ \ 2 // / pi\|
|\ (1 + x) / (1 + x)*sin|x + --||
\ \ 2 //
$$\left(- \frac{x^{2} \sin{\left(x \right)}}{\left(x + 1\right) \sin{\left(x + \frac{\pi}{2} \right)}} + \left(- \frac{x^{2}}{\left(x + 1\right)^{2}} + \frac{2 x}{x + 1}\right) \log{\left(\sin{\left(x + \frac{\pi}{2} \right)} \right)}\right) \sin^{\frac{x^{2}}{x + 1}}{\left(x + \frac{\pi}{2} \right)}$$
2
x / 2 / pi\\
----- |/ 2 \ x *cos|x - --||
1 + x || x 2*x | \ 2 /|
(cos(x)) *||- -------- + -----|*log(cos(x)) - --------------|
|| 2 1 + x| (1 + x)*cos(x)|
\\ (1 + x) / /
$$\left(- \frac{x^{2} \cos{\left(x - \frac{\pi}{2} \right)}}{\left(x + 1\right) \cos{\left(x \right)}} + \left(- \frac{x^{2}}{\left(x + 1\right)^{2}} + \frac{2 x}{x + 1}\right) \log{\left(\cos{\left(x \right)} \right)}\right) \cos^{\frac{x^{2}}{x + 1}}{\left(x \right)}$$
2
x
-----
1 + x
/ 2/x\\ / / 2/x\\ 2 /x\ \
|-1 + cot |-|| |/ 2 \ |-1 + cot |-|| 2*x *cot|-| |
| \2/| || x 2*x | | \2/| \2/ |
|------------| *||- -------- + -----|*log|------------| - ----------------------|
| 2/x\ | || 2 1 + x| | 2/x\ | / 2/x\\|
|1 + cot |-| | |\ (1 + x) / |1 + cot |-| | (1 + x)*|-1 + cot |-|||
\ \2/ / \ \ \2/ / \ \2///
$$\left(\frac{\cot^{2}{\left(\frac{x}{2} \right)} - 1}{\cot^{2}{\left(\frac{x}{2} \right)} + 1}\right)^{\frac{x^{2}}{x + 1}} \left(- \frac{2 x^{2} \cot{\left(\frac{x}{2} \right)}}{\left(x + 1\right) \left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)} + \left(- \frac{x^{2}}{\left(x + 1\right)^{2}} + \frac{2 x}{x + 1}\right) \log{\left(\frac{\cot^{2}{\left(\frac{x}{2} \right)} - 1}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} \right)}\right)$$
2
x
----- // 2 \ 2 \
x + 1 || -x 2*x | x *tan(x)|
(cos(x)) *||-------- + -----|*log(cos(x)) - ---------|
|| 2 x + 1| 1 + x |
\\(x + 1) / /
$$\left(- \frac{x^{2} \tan{\left(x \right)}}{x + 1} + \left(\frac{2 x}{x + 1} + \frac{\left(-1\right) x^{2}}{\left(x + 1\right)^{2}}\right) \log{\left(\cos{\left(x \right)} \right)}\right) \cos^{\frac{x^{2}}{x + 1}}{\left(x \right)}$$
2
x
-----
1 + x
/ 2/x\\ / / 2/x\\ \
|1 - tan |-|| |/ 2 \ |1 - tan |-|| 2 |
| \2/| || x 2*x | | \2/| x *tan(x)|
|-----------| *||- -------- + -----|*log|-----------| - ---------|
| 2/x\| || 2 1 + x| | 2/x\| 1 + x |
|1 + tan |-|| |\ (1 + x) / |1 + tan |-|| |
\ \2// \ \ \2// /
$$\left(\frac{1 - \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1}\right)^{\frac{x^{2}}{x + 1}} \left(- \frac{x^{2} \tan{\left(x \right)}}{x + 1} + \left(- \frac{x^{2}}{\left(x + 1\right)^{2}} + \frac{2 x}{x + 1}\right) \log{\left(\frac{1 - \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} \right)}\right)$$
2
x
----- // 2 \ 2 \
1 + x || x 2*x | x *tan(x)|
(cos(x)) *||- -------- + -----|*log(cos(x)) - ---------|
|| 2 1 + x| 1 + x |
\\ (1 + x) / /
$$\left(- \frac{x^{2} \tan{\left(x \right)}}{x + 1} + \left(- \frac{x^{2}}{\left(x + 1\right)^{2}} + \frac{2 x}{x + 1}\right) \log{\left(\cos{\left(x \right)} \right)}\right) \cos^{\frac{x^{2}}{x + 1}}{\left(x \right)}$$
2
x
-----
1 + x
/ 2/x\\ / / 2/x\\ 2 /x\ \
|1 - tan |-|| |/ 2 \ |1 - tan |-|| 2*x *tan|-| |
| \2/| || x 2*x | | \2/| \2/ |
|-----------| *||- -------- + -----|*log|-----------| - ---------------------|
| 2/x\| || 2 1 + x| | 2/x\| / 2/x\\|
|1 + tan |-|| |\ (1 + x) / |1 + tan |-|| (1 + x)*|1 - tan |-|||
\ \2// \ \ \2// \ \2///
$$\left(\frac{1 - \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1}\right)^{\frac{x^{2}}{x + 1}} \left(- \frac{2 x^{2} \tan{\left(\frac{x}{2} \right)}}{\left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right) \left(x + 1\right)} + \left(- \frac{x^{2}}{\left(x + 1\right)^{2}} + \frac{2 x}{x + 1}\right) \log{\left(\frac{1 - \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} \right)}\right)$$
2
x
-----
1 + x
/ 2/x\\ / / 2/x\\ \
|-1 + cot |-|| |/ 2 \ |-1 + cot |-|| 2 |
| \2/| || x 2*x | | \2/| x |
|------------| *||- -------- + -----|*log|------------| - --------------|
| 2/x\ | || 2 1 + x| | 2/x\ | (1 + x)*cot(x)|
|1 + cot |-| | |\ (1 + x) / |1 + cot |-| | |
\ \2/ / \ \ \2/ / /
$$\left(\frac{\cot^{2}{\left(\frac{x}{2} \right)} - 1}{\cot^{2}{\left(\frac{x}{2} \right)} + 1}\right)^{\frac{x^{2}}{x + 1}} \left(- \frac{x^{2}}{\left(x + 1\right) \cot{\left(x \right)}} + \left(- \frac{x^{2}}{\left(x + 1\right)^{2}} + \frac{2 x}{x + 1}\right) \log{\left(\frac{\cot^{2}{\left(\frac{x}{2} \right)} - 1}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} \right)}\right)$$
2
x
-----
1 + x // 2 \ 2 \
/ 1 \ || x 2*x | / 1 \ x *sec(x) |
|------| *||- -------- + -----|*log|------| - -------------------|
\sec(x)/ || 2 1 + x| \sec(x)/ / pi\|
|\ (1 + x) / (1 + x)*sec|x - --||
\ \ 2 //
$$\left(- \frac{x^{2} \sec{\left(x \right)}}{\left(x + 1\right) \sec{\left(x - \frac{\pi}{2} \right)}} + \left(- \frac{x^{2}}{\left(x + 1\right)^{2}} + \frac{2 x}{x + 1}\right) \log{\left(\frac{1}{\sec{\left(x \right)}} \right)}\right) \left(\frac{1}{\sec{\left(x \right)}}\right)^{\frac{x^{2}}{x + 1}}$$
2
x
----- // 2 \ 2 \
1 + x || x 2*x | x *sin(x) |
(cos(x)) *||- -------- + -----|*log(cos(x)) - --------------|
|| 2 1 + x| (1 + x)*cos(x)|
\\ (1 + x) / /
$$\left(- \frac{x^{2} \sin{\left(x \right)}}{\left(x + 1\right) \cos{\left(x \right)}} + \left(- \frac{x^{2}}{\left(x + 1\right)^{2}} + \frac{2 x}{x + 1}\right) \log{\left(\cos{\left(x \right)} \right)}\right) \cos^{\frac{x^{2}}{x + 1}}{\left(x \right)}$$
2
x
-----
1 + x // 2 \ 2 2 \
/ / pi\\ || x 2*x | / / pi\\ 2*x *sin (x) |
|sin|x + --|| *||- -------- + -----|*log|sin|x + --|| - ----------------|
\ \ 2 // || 2 1 + x| \ \ 2 // (1 + x)*sin(2*x)|
\\ (1 + x) / /
$$\left(- \frac{2 x^{2} \sin^{2}{\left(x \right)}}{\left(x + 1\right) \sin{\left(2 x \right)}} + \left(- \frac{x^{2}}{\left(x + 1\right)^{2}} + \frac{2 x}{x + 1}\right) \log{\left(\sin{\left(x + \frac{\pi}{2} \right)} \right)}\right) \sin^{\frac{x^{2}}{x + 1}}{\left(x + \frac{\pi}{2} \right)}$$
2
x
----- / 2 /pi \\
1 + x |/ 2 \ x *csc|-- - x||
/ 1 \ || x 2*x | / 1 \ \2 /|
|-----------| *||- -------- + -----|*log|-----------| - --------------|
| /pi \| || 2 1 + x| | /pi \| (1 + x)*csc(x)|
|csc|-- - x|| |\ (1 + x) / |csc|-- - x|| |
\ \2 // \ \ \2 // /
$$\left(- \frac{x^{2} \csc{\left(- x + \frac{\pi}{2} \right)}}{\left(x + 1\right) \csc{\left(x \right)}} + \left(- \frac{x^{2}}{\left(x + 1\right)^{2}} + \frac{2 x}{x + 1}\right) \log{\left(\frac{1}{\csc{\left(- x + \frac{\pi}{2} \right)}} \right)}\right) \left(\frac{1}{\csc{\left(- x + \frac{\pi}{2} \right)}}\right)^{\frac{x^{2}}{x + 1}}$$
2
x
-----
1 + x // 2 \ 2 \
/ 1 \ || x 2*x | / 1 \ x *sec(x) |
|------| *||- -------- + -----|*log|------| - --------------|
\sec(x)/ || 2 1 + x| \sec(x)/ (1 + x)*csc(x)|
\\ (1 + x) / /
$$\left(- \frac{x^{2} \sec{\left(x \right)}}{\left(x + 1\right) \csc{\left(x \right)}} + \left(- \frac{x^{2}}{\left(x + 1\right)^{2}} + \frac{2 x}{x + 1}\right) \log{\left(\frac{1}{\sec{\left(x \right)}} \right)}\right) \left(\frac{1}{\sec{\left(x \right)}}\right)^{\frac{x^{2}}{x + 1}}$$
(1/sec(x))^(x^2/(1 + x))*((-x^2/(1 + x)^2 + 2*x/(1 + x))*log(1/sec(x)) - x^2*sec(x)/((1 + x)*csc(x)))
Unión de expresiones racionales
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2
x
-----
1 + x
x*(cos(x)) *((2 + x)*cos(x)*log(cos(x)) - x*(1 + x)*sin(x))
---------------------------------------------------------------
2
(1 + x) *cos(x)
$$\frac{x \left(- x \left(x + 1\right) \sin{\left(x \right)} + \left(x + 2\right) \log{\left(\cos{\left(x \right)} \right)} \cos{\left(x \right)}\right) \cos^{\frac{x^{2}}{x + 1}}{\left(x \right)}}{\left(x + 1\right)^{2} \cos{\left(x \right)}}$$
x*cos(x)^(x^2/(1 + x))*((2 + x)*cos(x)*log(cos(x)) - x*(1 + x)*sin(x))/((1 + x)^2*cos(x))
Compilar la expresión
[src]
2
x
----- // 2 \ 2 \
x + 1 || x 2*x | x *sin(x) |
(cos(x)) *||- -------- + -----|*log(cos(x)) - --------------|
|| 2 1 + x| (1 + x)*cos(x)|
\\ (1 + x) / /
$$\left(- \frac{x^{2} \sin{\left(x \right)}}{\left(x + 1\right) \cos{\left(x \right)}} + \left(- \frac{x^{2}}{\left(x + 1\right)^{2}} + \frac{2 x}{x + 1}\right) \log{\left(\cos{\left(x \right)} \right)}\right) \cos^{\frac{x^{2}}{x + 1}}{\left(x \right)}$$
cos(x)^(x^2/(x + 1))*((-x^2/(1 + x)^2 + 2*x/(1 + x))*log(cos(x)) - x^2*sin(x)/((1 + x)*cos(x)))