Sr Examen
Simplificación de expresiones/
Descomposición en fracciones simples/
log(1+tan(x)^2)/(-4+4*tan(x)^2)-log(1+tan(x))/(-4+4*tan(x)^2)-log(-1+tan(x))/(-4+4*tan(x)^2)+tan(x)^2*log(1+tan(x))/(-4+4*tan(x)^2)+tan(x)^2*log(-1+tan(x))/(-4+4*tan(x)^2)-tan(x)^2*log(1+tan(x)^2)/(-4+4*tan(x)^2)-4*x*tan(x)/(-4+4*tan(x)^2)
¿Cómo vas a descomponer esta log(1+tan(x)^2)/(-4+4*tan(x)^2)-log(1+tan(x))/(-4+4*tan(x)^2)-log(-1+tan(x))/(-4+4*tan(x)^2)+tan(x)^2*log(1+tan(x))/(-4+4*tan(x)^2)+tan(x)^2*log(-1+tan(x))/(-4+4*tan(x)^2)-tan(x)^2*log(1+tan(x)^2)/(-4+4*tan(x)^2)-4*x*tan(x)/(-4+4*tan(x)^2) expresión en fracciones?
Solución
Ha introducido
[src]
/ 2 \ 2 2 2 / 2 \
log\1 + tan (x)/ log(1 + tan(x)) log(-1 + tan(x)) tan (x)*log(1 + tan(x)) tan (x)*log(-1 + tan(x)) tan (x)*log\1 + tan (x)/ 4*x*tan(x)
---------------- - --------------- - ---------------- + ----------------------- + ------------------------ - ------------------------ - --------------
2 2 2 2 2 2 2
-4 + 4*tan (x) -4 + 4*tan (x) -4 + 4*tan (x) -4 + 4*tan (x) -4 + 4*tan (x) -4 + 4*tan (x) -4 + 4*tan (x)
$$- \frac{4 x \tan{\left(x \right)}}{4 \tan^{2}{\left(x \right)} - 4} + \left(- \frac{\log{\left(\tan^{2}{\left(x \right)} + 1 \right)} \tan^{2}{\left(x \right)}}{4 \tan^{2}{\left(x \right)} - 4} + \left(\frac{\log{\left(\tan{\left(x \right)} - 1 \right)} \tan^{2}{\left(x \right)}}{4 \tan^{2}{\left(x \right)} - 4} + \left(\frac{\log{\left(\tan{\left(x \right)} + 1 \right)} \tan^{2}{\left(x \right)}}{4 \tan^{2}{\left(x \right)} - 4} + \left(\left(- \frac{\log{\left(\tan{\left(x \right)} + 1 \right)}}{4 \tan^{2}{\left(x \right)} - 4} + \frac{\log{\left(\tan^{2}{\left(x \right)} + 1 \right)}}{4 \tan^{2}{\left(x \right)} - 4}\right) - \frac{\log{\left(\tan{\left(x \right)} - 1 \right)}}{4 \tan^{2}{\left(x \right)} - 4}\right)\right)\right)\right)$$
log(1 + tan(x)^2)/(-4 + 4*tan(x)^2) - log(1 + tan(x))/(-4 + 4*tan(x)^2) - log(-1 + tan(x))/(-4 + 4*tan(x)^2) + (tan(x)^2*log(1 + tan(x)))/(-4 + 4*tan(x)^2) + (tan(x)^2*log(-1 + tan(x)))/(-4 + 4*tan(x)^2) - tan(x)^2*log(1 + tan(x)^2)/(-4 + 4*tan(x)^2) - (4*x)*tan(x)/(-4 + 4*tan(x)^2)
Simplificación general
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2 2 2 / 1 \ / 1 \
-log(1 + tan(x)) - log(-1 + tan(x)) + tan (x)*log(1 + tan(x)) + tan (x)*log(-1 + tan(x)) - tan (x)*log|-------| - 4*x*tan(x) + log|-------|
| 2 | | 2 |
\cos (x)/ \cos (x)/
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/ 2 \
4*\-1 + tan (x)/
$$\frac{- 4 x \tan{\left(x \right)} + \log{\left(\tan{\left(x \right)} - 1 \right)} \tan^{2}{\left(x \right)} - \log{\left(\tan{\left(x \right)} - 1 \right)} + \log{\left(\tan{\left(x \right)} + 1 \right)} \tan^{2}{\left(x \right)} - \log{\left(\tan{\left(x \right)} + 1 \right)} - \log{\left(\frac{1}{\cos^{2}{\left(x \right)}} \right)} \tan^{2}{\left(x \right)} + \log{\left(\frac{1}{\cos^{2}{\left(x \right)}} \right)}}{4 \left(\tan^{2}{\left(x \right)} - 1\right)}$$
(-log(1 + tan(x)) - log(-1 + tan(x)) + tan(x)^2*log(1 + tan(x)) + tan(x)^2*log(-1 + tan(x)) - tan(x)^2*log(cos(x)^(-2)) - 4*x*tan(x) + log(cos(x)^(-2)))/(4*(-1 + tan(x)^2))
Respuesta numérica
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log(1 + tan(x)^2)/(-4.0 + 4.0*tan(x)^2) - log(1 + tan(x))/(-4.0 + 4.0*tan(x)^2) - log(-1 + tan(x))/(-4.0 + 4.0*tan(x)^2) + tan(x)^2*log(1 + tan(x))/(-4.0 + 4.0*tan(x)^2) + tan(x)^2*log(-1 + tan(x))/(-4.0 + 4.0*tan(x)^2) - tan(x)^2*log(1 + tan(x)^2)/(-4.0 + 4.0*tan(x)^2) - 4.0*x*tan(x)/(-4.0 + 4.0*tan(x)^2)
log(1 + tan(x)^2)/(-4.0 + 4.0*tan(x)^2) - log(1 + tan(x))/(-4.0 + 4.0*tan(x)^2) - log(-1 + tan(x))/(-4.0 + 4.0*tan(x)^2) + tan(x)^2*log(1 + tan(x))/(-4.0 + 4.0*tan(x)^2) + tan(x)^2*log(-1 + tan(x))/(-4.0 + 4.0*tan(x)^2) - tan(x)^2*log(1 + tan(x)^2)/(-4.0 + 4.0*tan(x)^2) - 4.0*x*tan(x)/(-4.0 + 4.0*tan(x)^2)
Denominador racional
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2 2 2 / 2 \ / 2 \
-log(1 + tan(x)) - log(-1 + tan(x)) + tan (x)*log(1 + tan(x)) + tan (x)*log(-1 + tan(x)) - tan (x)*log\1 + tan (x)/ - 4*x*tan(x) + log\1 + tan (x)/
---------------------------------------------------------------------------------------------------------------------------------------------------
2
-4 + 4*tan (x)
$$\frac{- 4 x \tan{\left(x \right)} + \log{\left(\tan{\left(x \right)} - 1 \right)} \tan^{2}{\left(x \right)} - \log{\left(\tan{\left(x \right)} - 1 \right)} + \log{\left(\tan{\left(x \right)} + 1 \right)} \tan^{2}{\left(x \right)} - \log{\left(\tan{\left(x \right)} + 1 \right)} - \log{\left(\tan^{2}{\left(x \right)} + 1 \right)} \tan^{2}{\left(x \right)} + \log{\left(\tan^{2}{\left(x \right)} + 1 \right)}}{4 \tan^{2}{\left(x \right)} - 4}$$
(-log(1 + tan(x)) - log(-1 + tan(x)) + tan(x)^2*log(1 + tan(x)) + tan(x)^2*log(-1 + tan(x)) - tan(x)^2*log(1 + tan(x)^2) - 4*x*tan(x) + log(1 + tan(x)^2))/(-4 + 4*tan(x)^2)
Unión de expresiones racionales
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2 2 2 / 2 \ / 2 \
-log(1 + tan(x)) - log(-1 + tan(x)) + tan (x)*log(1 + tan(x)) + tan (x)*log(-1 + tan(x)) - tan (x)*log\1 + tan (x)/ - 4*x*tan(x) + log\1 + tan (x)/
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/ 2 \
4*\-1 + tan (x)/
$$\frac{- 4 x \tan{\left(x \right)} + \log{\left(\tan{\left(x \right)} - 1 \right)} \tan^{2}{\left(x \right)} - \log{\left(\tan{\left(x \right)} - 1 \right)} + \log{\left(\tan{\left(x \right)} + 1 \right)} \tan^{2}{\left(x \right)} - \log{\left(\tan{\left(x \right)} + 1 \right)} - \log{\left(\tan^{2}{\left(x \right)} + 1 \right)} \tan^{2}{\left(x \right)} + \log{\left(\tan^{2}{\left(x \right)} + 1 \right)}}{4 \left(\tan^{2}{\left(x \right)} - 1\right)}$$
(-log(1 + tan(x)) - log(-1 + tan(x)) + tan(x)^2*log(1 + tan(x)) + tan(x)^2*log(-1 + tan(x)) - tan(x)^2*log(1 + tan(x)^2) - 4*x*tan(x) + log(1 + tan(x)^2))/(4*(-1 + tan(x)^2))
Denominador común
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/ 2 \
log\1 + tan (x)/ log(1 + tan(x)) log(-1 + tan(x)) x*tan(x)
- ---------------- + --------------- + ---------------- - ------------
4 4 4 2
-1 + tan (x)
$$- \frac{x \tan{\left(x \right)}}{\tan^{2}{\left(x \right)} - 1} + \frac{\log{\left(\tan{\left(x \right)} - 1 \right)}}{4} + \frac{\log{\left(\tan{\left(x \right)} + 1 \right)}}{4} - \frac{\log{\left(\tan^{2}{\left(x \right)} + 1 \right)}}{4}$$
-log(1 + tan(x)^2)/4 + log(1 + tan(x))/4 + log(-1 + tan(x))/4 - x*tan(x)/(-1 + tan(x)^2)
Combinatoria
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/ / 2 \ 2 / 2 \ 2 2 \
-\- log\1 + tan (x)/ + tan (x)*log\1 + tan (x)/ - tan (x)*log(1 + tan(x)) - tan (x)*log(-1 + tan(x)) + 4*x*tan(x) + log(1 + tan(x)) + log(-1 + tan(x))/
--------------------------------------------------------------------------------------------------------------------------------------------------------
4*(1 + tan(x))*(-1 + tan(x))
$$- \frac{4 x \tan{\left(x \right)} - \log{\left(\tan{\left(x \right)} - 1 \right)} \tan^{2}{\left(x \right)} + \log{\left(\tan{\left(x \right)} - 1 \right)} - \log{\left(\tan{\left(x \right)} + 1 \right)} \tan^{2}{\left(x \right)} + \log{\left(\tan{\left(x \right)} + 1 \right)} + \log{\left(\tan^{2}{\left(x \right)} + 1 \right)} \tan^{2}{\left(x \right)} - \log{\left(\tan^{2}{\left(x \right)} + 1 \right)}}{4 \left(\tan{\left(x \right)} - 1\right) \left(\tan{\left(x \right)} + 1\right)}$$
-(-log(1 + tan(x)^2) + tan(x)^2*log(1 + tan(x)^2) - tan(x)^2*log(1 + tan(x)) - tan(x)^2*log(-1 + tan(x)) + 4*x*tan(x) + log(1 + tan(x)) + log(-1 + tan(x)))/(4*(1 + tan(x))*(-1 + tan(x)))
Potencias
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/ 2 \ 2 2 2 / 2 \
log\1 + tan (x)/ log(1 + tan(x)) log(-1 + tan(x)) tan (x)*log(1 + tan(x)) tan (x)*log(-1 + tan(x)) tan (x)*log\1 + tan (x)/ 4*x*tan(x)
---------------- - --------------- - ---------------- + ----------------------- + ------------------------ - ------------------------ - --------------
2 2 2 2 2 2 2
-4 + 4*tan (x) -4 + 4*tan (x) -4 + 4*tan (x) -4 + 4*tan (x) -4 + 4*tan (x) -4 + 4*tan (x) -4 + 4*tan (x)
$$- \frac{4 x \tan{\left(x \right)}}{4 \tan^{2}{\left(x \right)} - 4} + \frac{\log{\left(\tan{\left(x \right)} - 1 \right)} \tan^{2}{\left(x \right)}}{4 \tan^{2}{\left(x \right)} - 4} - \frac{\log{\left(\tan{\left(x \right)} - 1 \right)}}{4 \tan^{2}{\left(x \right)} - 4} + \frac{\log{\left(\tan{\left(x \right)} + 1 \right)} \tan^{2}{\left(x \right)}}{4 \tan^{2}{\left(x \right)} - 4} - \frac{\log{\left(\tan{\left(x \right)} + 1 \right)}}{4 \tan^{2}{\left(x \right)} - 4} - \frac{\log{\left(\tan^{2}{\left(x \right)} + 1 \right)} \tan^{2}{\left(x \right)}}{4 \tan^{2}{\left(x \right)} - 4} + \frac{\log{\left(\tan^{2}{\left(x \right)} + 1 \right)}}{4 \tan^{2}{\left(x \right)} - 4}$$
/ 2\ / 2\
| / I*x -I*x\ | 2 | / I*x -I*x\ |
| \- e + e / | / / I*x -I*x\\ / / I*x -I*x\\ / I*x -I*x\ | \- e + e / | 2 / / I*x -I*x\\ 2 / / I*x -I*x\\
log|1 - -----------------| | I*\- e + e /| | I*\- e + e /| \- e + e / *log|1 - -----------------| / I*x -I*x\ | I*\- e + e /| / I*x -I*x\ | I*\- e + e /|
| 2 | log|1 + ------------------| log|-1 + ------------------| | 2 | \- e + e / *log|1 + ------------------| \- e + e / *log|-1 + ------------------|
| / I*x -I*x\ | | I*x -I*x | | I*x -I*x | | / I*x -I*x\ | | I*x -I*x | | I*x -I*x | / I*x -I*x\
\ \e + e / / \ e + e / \ e + e / \ \e + e / / \ e + e / \ e + e / 4*I*x*\- e + e /
-------------------------- - --------------------------- - ---------------------------- + -------------------------------------------- - --------------------------------------------- - ---------------------------------------------- - -----------------------------------------
2 2 2 / 2\ / 2\ / 2\ / 2\
/ I*x -I*x\ / I*x -I*x\ / I*x -I*x\ | / I*x -I*x\ | 2 | / I*x -I*x\ | 2 | / I*x -I*x\ | 2 | / I*x -I*x\ |
4*\- e + e / 4*\- e + e / 4*\- e + e / | 4*\- e + e / | / I*x -I*x\ | 4*\- e + e / | / I*x -I*x\ | 4*\- e + e / | / I*x -I*x\ | 4*\- e + e / | / I*x -I*x\
-4 - ------------------- -4 - ------------------- -4 - ------------------- |-4 - -------------------|*\e + e / |-4 - -------------------|*\e + e / |-4 - -------------------|*\e + e / |-4 - -------------------|*\e + e /
2 2 2 | 2 | | 2 | | 2 | | 2 |
/ I*x -I*x\ / I*x -I*x\ / I*x -I*x\ | / I*x -I*x\ | | / I*x -I*x\ | | / I*x -I*x\ | | / I*x -I*x\ |
\e + e / \e + e / \e + e / \ \e + e / / \ \e + e / / \ \e + e / / \ \e + e / /
$$- \frac{4 i x \left(- e^{i x} + e^{- i x}\right)}{\left(- \frac{4 \left(- e^{i x} + e^{- i x}\right)^{2}}{\left(e^{i x} + e^{- i x}\right)^{2}} - 4\right) \left(e^{i x} + e^{- i x}\right)} + \frac{\left(- e^{i x} + e^{- i x}\right)^{2} \log{\left(- \frac{\left(- e^{i x} + e^{- i x}\right)^{2}}{\left(e^{i x} + e^{- i x}\right)^{2}} + 1 \right)}}{\left(- \frac{4 \left(- e^{i x} + e^{- i x}\right)^{2}}{\left(e^{i x} + e^{- i x}\right)^{2}} - 4\right) \left(e^{i x} + e^{- i x}\right)^{2}} - \frac{\left(- e^{i x} + e^{- i x}\right)^{2} \log{\left(\frac{i \left(- e^{i x} + e^{- i x}\right)}{e^{i x} + e^{- i x}} - 1 \right)}}{\left(- \frac{4 \left(- e^{i x} + e^{- i x}\right)^{2}}{\left(e^{i x} + e^{- i x}\right)^{2}} - 4\right) \left(e^{i x} + e^{- i x}\right)^{2}} - \frac{\left(- e^{i x} + e^{- i x}\right)^{2} \log{\left(\frac{i \left(- e^{i x} + e^{- i x}\right)}{e^{i x} + e^{- i x}} + 1 \right)}}{\left(- \frac{4 \left(- e^{i x} + e^{- i x}\right)^{2}}{\left(e^{i x} + e^{- i x}\right)^{2}} - 4\right) \left(e^{i x} + e^{- i x}\right)^{2}} + \frac{\log{\left(- \frac{\left(- e^{i x} + e^{- i x}\right)^{2}}{\left(e^{i x} + e^{- i x}\right)^{2}} + 1 \right)}}{- \frac{4 \left(- e^{i x} + e^{- i x}\right)^{2}}{\left(e^{i x} + e^{- i x}\right)^{2}} - 4} - \frac{\log{\left(\frac{i \left(- e^{i x} + e^{- i x}\right)}{e^{i x} + e^{- i x}} - 1 \right)}}{- \frac{4 \left(- e^{i x} + e^{- i x}\right)^{2}}{\left(e^{i x} + e^{- i x}\right)^{2}} - 4} - \frac{\log{\left(\frac{i \left(- e^{i x} + e^{- i x}\right)}{e^{i x} + e^{- i x}} + 1 \right)}}{- \frac{4 \left(- e^{i x} + e^{- i x}\right)^{2}}{\left(e^{i x} + e^{- i x}\right)^{2}} - 4}$$
log(1 - (-exp(i*x) + exp(-i*x))^2/(exp(i*x) + exp(-i*x))^2)/(-4 - 4*(-exp(i*x) + exp(-i*x))^2/(exp(i*x) + exp(-i*x))^2) - log(1 + i*(-exp(i*x) + exp(-i*x))/(exp(i*x) + exp(-i*x)))/(-4 - 4*(-exp(i*x) + exp(-i*x))^2/(exp(i*x) + exp(-i*x))^2) - log(-1 + i*(-exp(i*x) + exp(-i*x))/(exp(i*x) + exp(-i*x)))/(-4 - 4*(-exp(i*x) + exp(-i*x))^2/(exp(i*x) + exp(-i*x))^2) + (-exp(i*x) + exp(-i*x))^2*log(1 - (-exp(i*x) + exp(-i*x))^2/(exp(i*x) + exp(-i*x))^2)/((-4 - 4*(-exp(i*x) + exp(-i*x))^2/(exp(i*x) + exp(-i*x))^2)*(exp(i*x) + exp(-i*x))^2) - (-exp(i*x) + exp(-i*x))^2*log(1 + i*(-exp(i*x) + exp(-i*x))/(exp(i*x) + exp(-i*x)))/((-4 - 4*(-exp(i*x) + exp(-i*x))^2/(exp(i*x) + exp(-i*x))^2)*(exp(i*x) + exp(-i*x))^2) - (-exp(i*x) + exp(-i*x))^2*log(-1 + i*(-exp(i*x) + exp(-i*x))/(exp(i*x) + exp(-i*x)))/((-4 - 4*(-exp(i*x) + exp(-i*x))^2/(exp(i*x) + exp(-i*x))^2)*(exp(i*x) + exp(-i*x))^2) - 4*i*x*(-exp(i*x) + exp(-i*x))/((-4 - 4*(-exp(i*x) + exp(-i*x))^2/(exp(i*x) + exp(-i*x))^2)*(exp(i*x) + exp(-i*x)))
Compilar la expresión
[src]
/ 2 \ 2 2 2 / 2 \
log\1 + tan (x)/ log(1 + tan(x)) log(-1 + tan(x)) tan (x)*log(1 + tan(x)) tan (x)*log(-1 + tan(x)) tan (x)*log\1 + tan (x)/ 4*x*tan(x)
---------------- - --------------- - ---------------- + ----------------------- + ------------------------ - ------------------------ - --------------
2 2 2 2 2 2 2
-4 + 4*tan (x) -4 + 4*tan (x) -4 + 4*tan (x) -4 + 4*tan (x) -4 + 4*tan (x) -4 + 4*tan (x) -4 + 4*tan (x)
$$- \frac{4 x \tan{\left(x \right)}}{4 \tan^{2}{\left(x \right)} - 4} + \frac{\log{\left(\tan{\left(x \right)} - 1 \right)} \tan^{2}{\left(x \right)}}{4 \tan^{2}{\left(x \right)} - 4} - \frac{\log{\left(\tan{\left(x \right)} - 1 \right)}}{4 \tan^{2}{\left(x \right)} - 4} + \frac{\log{\left(\tan{\left(x \right)} + 1 \right)} \tan^{2}{\left(x \right)}}{4 \tan^{2}{\left(x \right)} - 4} - \frac{\log{\left(\tan{\left(x \right)} + 1 \right)}}{4 \tan^{2}{\left(x \right)} - 4} - \frac{\log{\left(\tan^{2}{\left(x \right)} + 1 \right)} \tan^{2}{\left(x \right)}}{4 \tan^{2}{\left(x \right)} - 4} + \frac{\log{\left(\tan^{2}{\left(x \right)} + 1 \right)}}{4 \tan^{2}{\left(x \right)} - 4}$$
log(1 + tan(x)^2)/(-4 + 4*tan(x)^2) - log(1 + tan(x))/(-4 + 4*tan(x)^2) - log(-1 + tan(x))/(-4 + 4*tan(x)^2) + tan(x)^2*log(1 + tan(x))/(-4 + 4*tan(x)^2) + tan(x)^2*log(-1 + tan(x))/(-4 + 4*tan(x)^2) - tan(x)^2*log(1 + tan(x)^2)/(-4 + 4*tan(x)^2) - 4*x*tan(x)/(-4 + 4*tan(x)^2)