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