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