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