Sr Examen
Simplificación de expresiones/
Descomposición en fracciones simples/
log(1-tan(x/2)^2+2*tan(x/2))/(4+4*tan(x/2)^4+8*tan(x/2)^2)-log(1+tan(x/2)^2)/(4+4*tan(x/2)^4+8*tan(x/2)^2)-2*tan(x/2)/(4+4*tan(x/2)^4+8*tan(x/2)^2)+2*tan(x/2)^3/(4+4*tan(x/2)^4+8*tan(x/2)^2)+4*tan(x/2)^2/(4+4*tan(x/2)^4+8*tan(x/2)^2)+tan(x/2)^4*log(1-tan(x/2)^2+2*tan(x/2))/(4+4*tan(x/2)^4+8*tan(x/2)^2)-tan(x/2)^4*log(1+tan(x/2)^2)/(4+4*tan(x/2)^4+8*tan(x/2)^2)-2*tan(x/2)^2*log(1+tan(x/2)^2)/(4+4*tan(x/2)^4+8*tan(x/2)^2)+2*tan(x/2)^2*log(1-tan(x/2)^2+2*tan(x/2))/(4+4*tan(x/2)^4+8*tan(x/2)^2)
¿Cómo vas a descomponer esta log(1-tan(x/2)^2+2*tan(x/2))/(4+4*tan(x/2)^4+8*tan(x/2)^2)-log(1+tan(x/2)^2)/(4+4*tan(x/2)^4+8*tan(x/2)^2)-2*tan(x/2)/(4+4*tan(x/2)^4+8*tan(x/2)^2)+2*tan(x/2)^3/(4+4*tan(x/2)^4+8*tan(x/2)^2)+4*tan(x/2)^2/(4+4*tan(x/2)^4+8*tan(x/2)^2)+tan(x/2)^4*log(1-tan(x/2)^2+2*tan(x/2))/(4+4*tan(x/2)^4+8*tan(x/2)^2)-tan(x/2)^4*log(1+tan(x/2)^2)/(4+4*tan(x/2)^4+8*tan(x/2)^2)-2*tan(x/2)^2*log(1+tan(x/2)^2)/(4+4*tan(x/2)^4+8*tan(x/2)^2)+2*tan(x/2)^2*log(1-tan(x/2)^2+2*tan(x/2))/(4+4*tan(x/2)^4+8*tan(x/2)^2) expresión en fracciones?
Solución
Ha introducido
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/ 2/x\ /x\\ / 2/x\\ /x\ 3/x\ 2/x\ 4/x\ / 2/x\ /x\\ 4/x\ / 2/x\\ 2/x\ / 2/x\\ 2/x\ / 2/x\ /x\\
log|1 - tan |-| + 2*tan|-|| log|1 + tan |-|| 2*tan|-| 2*tan |-| 4*tan |-| tan |-|*log|1 - tan |-| + 2*tan|-|| tan |-|*log|1 + tan |-|| 2*tan |-|*log|1 + tan |-|| 2*tan |-|*log|1 - tan |-| + 2*tan|-||
\ \2/ \2// \ \2// \2/ \2/ \2/ \2/ \ \2/ \2// \2/ \ \2// \2/ \ \2// \2/ \ \2/ \2//
--------------------------- - ------------------------- - ------------------------- + ------------------------- + ------------------------- + ----------------------------------- - ------------------------- - -------------------------- + -------------------------------------
4/x\ 2/x\ 4/x\ 2/x\ 4/x\ 2/x\ 4/x\ 2/x\ 4/x\ 2/x\ 4/x\ 2/x\ 4/x\ 2/x\ 4/x\ 2/x\ 4/x\ 2/x\
4 + 4*tan |-| + 8*tan |-| 4 + 4*tan |-| + 8*tan |-| 4 + 4*tan |-| + 8*tan |-| 4 + 4*tan |-| + 8*tan |-| 4 + 4*tan |-| + 8*tan |-| 4 + 4*tan |-| + 8*tan |-| 4 + 4*tan |-| + 8*tan |-| 4 + 4*tan |-| + 8*tan |-| 4 + 4*tan |-| + 8*tan |-|
\2/ \2/ \2/ \2/ \2/ \2/ \2/ \2/ \2/ \2/ \2/ \2/ \2/ \2/ \2/ \2/ \2/ \2/
$$\frac{\log{\left(\left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right) + 2 \tan{\left(\frac{x}{2} \right)} \right)} 2 \tan^{2}{\left(\frac{x}{2} \right)}}{\left(4 \tan^{4}{\left(\frac{x}{2} \right)} + 4\right) + 8 \tan^{2}{\left(\frac{x}{2} \right)}} + \left(- \frac{\log{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1 \right)} 2 \tan^{2}{\left(\frac{x}{2} \right)}}{\left(4 \tan^{4}{\left(\frac{x}{2} \right)} + 4\right) + 8 \tan^{2}{\left(\frac{x}{2} \right)}} + \left(- \frac{\log{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1 \right)} \tan^{4}{\left(\frac{x}{2} \right)}}{\left(4 \tan^{4}{\left(\frac{x}{2} \right)} + 4\right) + 8 \tan^{2}{\left(\frac{x}{2} \right)}} + \left(\frac{\log{\left(\left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right) + 2 \tan{\left(\frac{x}{2} \right)} \right)} \tan^{4}{\left(\frac{x}{2} \right)}}{\left(4 \tan^{4}{\left(\frac{x}{2} \right)} + 4\right) + 8 \tan^{2}{\left(\frac{x}{2} \right)}} + \left(\left(\left(\left(\frac{\log{\left(\left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right) + 2 \tan{\left(\frac{x}{2} \right)} \right)}}{\left(4 \tan^{4}{\left(\frac{x}{2} \right)} + 4\right) + 8 \tan^{2}{\left(\frac{x}{2} \right)}} - \frac{\log{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1 \right)}}{\left(4 \tan^{4}{\left(\frac{x}{2} \right)} + 4\right) + 8 \tan^{2}{\left(\frac{x}{2} \right)}}\right) - \frac{2 \tan{\left(\frac{x}{2} \right)}}{\left(4 \tan^{4}{\left(\frac{x}{2} \right)} + 4\right) + 8 \tan^{2}{\left(\frac{x}{2} \right)}}\right) + \frac{2 \tan^{3}{\left(\frac{x}{2} \right)}}{\left(4 \tan^{4}{\left(\frac{x}{2} \right)} + 4\right) + 8 \tan^{2}{\left(\frac{x}{2} \right)}}\right) + \frac{4 \tan^{2}{\left(\frac{x}{2} \right)}}{\left(4 \tan^{4}{\left(\frac{x}{2} \right)} + 4\right) + 8 \tan^{2}{\left(\frac{x}{2} \right)}}\right)\right)\right)\right)$$
log(1 - tan(x/2)^2 + 2*tan(x/2))/(4 + 4*tan(x/2)^4 + 8*tan(x/2)^2) - log(1 + tan(x/2)^2)/(4 + 4*tan(x/2)^4 + 8*tan(x/2)^2) - 2*tan(x/2)/(4 + 4*tan(x/2)^4 + 8*tan(x/2)^2) + (2*tan(x/2)^3)/(4 + 4*tan(x/2)^4 + 8*tan(x/2)^2) + (4*tan(x/2)^2)/(4 + 4*tan(x/2)^4 + 8*tan(x/2)^2) + (tan(x/2)^4*log(1 - tan(x/2)^2 + 2*tan(x/2)))/(4 + 4*tan(x/2)^4 + 8*tan(x/2)^2) - tan(x/2)^4*log(1 + tan(x/2)^2)/(4 + 4*tan(x/2)^4 + 8*tan(x/2)^2) - (2*tan(x/2)^2)*log(1 + tan(x/2)^2)/(4 + 4*tan(x/2)^4 + 8*tan(x/2)^2) + ((2*tan(x/2)^2)*log(1 - tan(x/2)^2 + 2*tan(x/2)))/(4 + 4*tan(x/2)^4 + 8*tan(x/2)^2)
Simplificación general
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/ 1 \ / 2/x\ /x\\
2 log|----------| log|1 - tan |-| + 2*tan|-||
1 cos (x) log(2) \1 + cos(x)/ sin(2*x) \ \2/ \2//
- - ------- - ------ - --------------- - -------- + ---------------------------
4 4 4 4 8 4
$$- \frac{\log{\left(\frac{1}{\cos{\left(x \right)} + 1} \right)}}{4} + \frac{\log{\left(- \tan^{2}{\left(\frac{x}{2} \right)} + 2 \tan{\left(\frac{x}{2} \right)} + 1 \right)}}{4} - \frac{\sin{\left(2 x \right)}}{8} - \frac{\cos^{2}{\left(x \right)}}{4} - \frac{\log{\left(2 \right)}}{4} + \frac{1}{4}$$
1/4 - cos(x)^2/4 - log(2)/4 - log(1/(1 + cos(x)))/4 - sin(2*x)/8 + log(1 - tan(x/2)^2 + 2*tan(x/2))/4
Respuesta numérica
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log(1 - tan(x/2)^2 + 2*tan(x/2))/(4.0 + 4.0*tan(x/2)^4 + 8.0*tan(x/2)^2) - log(1 + tan(x/2)^2)/(4.0 + 4.0*tan(x/2)^4 + 8.0*tan(x/2)^2) + 2.0*tan(x/2)^3/(4.0 + 4.0*tan(x/2)^4 + 8.0*tan(x/2)^2) + 4.0*tan(x/2)^2/(4.0 + 4.0*tan(x/2)^4 + 8.0*tan(x/2)^2) - 2.0*tan(x/2)/(4.0 + 4.0*tan(x/2)^4 + 8.0*tan(x/2)^2) + tan(x/2)^4*log(1 - tan(x/2)^2 + 2*tan(x/2))/(4.0 + 4.0*tan(x/2)^4 + 8.0*tan(x/2)^2) - tan(x/2)^4*log(1 + tan(x/2)^2)/(4.0 + 4.0*tan(x/2)^4 + 8.0*tan(x/2)^2) + 2.0*tan(x/2)^2*log(1 - tan(x/2)^2 + 2*tan(x/2))/(4.0 + 4.0*tan(x/2)^4 + 8.0*tan(x/2)^2) - 2.0*tan(x/2)^2*log(1 + tan(x/2)^2)/(4.0 + 4.0*tan(x/2)^4 + 8.0*tan(x/2)^2)
log(1 - tan(x/2)^2 + 2*tan(x/2))/(4.0 + 4.0*tan(x/2)^4 + 8.0*tan(x/2)^2) - log(1 + tan(x/2)^2)/(4.0 + 4.0*tan(x/2)^4 + 8.0*tan(x/2)^2) + 2.0*tan(x/2)^3/(4.0 + 4.0*tan(x/2)^4 + 8.0*tan(x/2)^2) + 4.0*tan(x/2)^2/(4.0 + 4.0*tan(x/2)^4 + 8.0*tan(x/2)^2) - 2.0*tan(x/2)/(4.0 + 4.0*tan(x/2)^4 + 8.0*tan(x/2)^2) + tan(x/2)^4*log(1 - tan(x/2)^2 + 2*tan(x/2))/(4.0 + 4.0*tan(x/2)^4 + 8.0*tan(x/2)^2) - tan(x/2)^4*log(1 + tan(x/2)^2)/(4.0 + 4.0*tan(x/2)^4 + 8.0*tan(x/2)^2) + 2.0*tan(x/2)^2*log(1 - tan(x/2)^2 + 2*tan(x/2))/(4.0 + 4.0*tan(x/2)^4 + 8.0*tan(x/2)^2) - 2.0*tan(x/2)^2*log(1 + tan(x/2)^2)/(4.0 + 4.0*tan(x/2)^4 + 8.0*tan(x/2)^2)
Denominador común
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/ 2/x\\ / 2/x\ /x\\ 3/x\ /x\ 2/x\
log|1 + tan |-|| log|1 - tan |-| + 2*tan|-|| tan |-| - tan|-| + 2*tan |-|
\ \2// \ \2/ \2// \2/ \2/ \2/
- ---------------- + --------------------------- + ----------------------------
4 4 4/x\ 2/x\
2 + 2*tan |-| + 4*tan |-|
\2/ \2/
$$\frac{\tan^{3}{\left(\frac{x}{2} \right)} + 2 \tan^{2}{\left(\frac{x}{2} \right)} - \tan{\left(\frac{x}{2} \right)}}{2 \tan^{4}{\left(\frac{x}{2} \right)} + 4 \tan^{2}{\left(\frac{x}{2} \right)} + 2} - \frac{\log{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1 \right)}}{4} + \frac{\log{\left(- \tan^{2}{\left(\frac{x}{2} \right)} + 2 \tan{\left(\frac{x}{2} \right)} + 1 \right)}}{4}$$
-log(1 + tan(x/2)^2)/4 + log(1 - tan(x/2)^2 + 2*tan(x/2))/4 + (tan(x/2)^3 - tan(x/2) + 2*tan(x/2)^2)/(2 + 2*tan(x/2)^4 + 4*tan(x/2)^2)
Denominador racional
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/ 2/x\\ /x\ 3/x\ 2/x\ 4/x\ / 2/x\ /x\\ 4/x\ / 2/x\\ 2/x\ / 2/x\\ 2/x\ / 2/x\ /x\\ / 2/x\ /x\\
- log|1 + tan |-|| - 2*tan|-| + 2*tan |-| + 4*tan |-| + tan |-|*log|1 - tan |-| + 2*tan|-|| - tan |-|*log|1 + tan |-|| - 2*tan |-|*log|1 + tan |-|| + 2*tan |-|*log|1 - tan |-| + 2*tan|-|| + log|1 - tan |-| + 2*tan|-||
\ \2// \2/ \2/ \2/ \2/ \ \2/ \2// \2/ \ \2// \2/ \ \2// \2/ \ \2/ \2// \ \2/ \2//
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
4/x\ 2/x\
4 + 4*tan |-| + 8*tan |-|
\2/ \2/
$$\frac{- \log{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1 \right)} \tan^{4}{\left(\frac{x}{2} \right)} - 2 \log{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1 \right)} \tan^{2}{\left(\frac{x}{2} \right)} - \log{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1 \right)} + \log{\left(- \tan^{2}{\left(\frac{x}{2} \right)} + 2 \tan{\left(\frac{x}{2} \right)} + 1 \right)} \tan^{4}{\left(\frac{x}{2} \right)} + 2 \log{\left(- \tan^{2}{\left(\frac{x}{2} \right)} + 2 \tan{\left(\frac{x}{2} \right)} + 1 \right)} \tan^{2}{\left(\frac{x}{2} \right)} + \log{\left(- \tan^{2}{\left(\frac{x}{2} \right)} + 2 \tan{\left(\frac{x}{2} \right)} + 1 \right)} + 2 \tan^{3}{\left(\frac{x}{2} \right)} + 4 \tan^{2}{\left(\frac{x}{2} \right)} - 2 \tan{\left(\frac{x}{2} \right)}}{4 \tan^{4}{\left(\frac{x}{2} \right)} + 8 \tan^{2}{\left(\frac{x}{2} \right)} + 4}$$
(-log(1 + tan(x/2)^2) - 2*tan(x/2) + 2*tan(x/2)^3 + 4*tan(x/2)^2 + tan(x/2)^4*log(1 - tan(x/2)^2 + 2*tan(x/2)) - tan(x/2)^4*log(1 + tan(x/2)^2) - 2*tan(x/2)^2*log(1 + tan(x/2)^2) + 2*tan(x/2)^2*log(1 - tan(x/2)^2 + 2*tan(x/2)) + log(1 - tan(x/2)^2 + 2*tan(x/2)))/(4 + 4*tan(x/2)^4 + 8*tan(x/2)^2)
Unión de expresiones racionales
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/ 2/x\\ /x\ 3/x\ 2/x\ 4/x\ / 2/x\ /x\\ 4/x\ / 2/x\\ 2/x\ / 2/x\\ 2/x\ / 2/x\ /x\\ / 2/x\ /x\\
- log|1 + tan |-|| - 2*tan|-| + 2*tan |-| + 4*tan |-| + tan |-|*log|1 - tan |-| + 2*tan|-|| - tan |-|*log|1 + tan |-|| - 2*tan |-|*log|1 + tan |-|| + 2*tan |-|*log|1 - tan |-| + 2*tan|-|| + log|1 - tan |-| + 2*tan|-||
\ \2// \2/ \2/ \2/ \2/ \ \2/ \2// \2/ \ \2// \2/ \ \2// \2/ \ \2/ \2// \ \2/ \2//
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
/ 4/x\ 2/x\\
4*|1 + tan |-| + 2*tan |-||
\ \2/ \2//
$$\frac{- \log{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1 \right)} \tan^{4}{\left(\frac{x}{2} \right)} - 2 \log{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1 \right)} \tan^{2}{\left(\frac{x}{2} \right)} - \log{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1 \right)} + \log{\left(- \tan^{2}{\left(\frac{x}{2} \right)} + 2 \tan{\left(\frac{x}{2} \right)} + 1 \right)} \tan^{4}{\left(\frac{x}{2} \right)} + 2 \log{\left(- \tan^{2}{\left(\frac{x}{2} \right)} + 2 \tan{\left(\frac{x}{2} \right)} + 1 \right)} \tan^{2}{\left(\frac{x}{2} \right)} + \log{\left(- \tan^{2}{\left(\frac{x}{2} \right)} + 2 \tan{\left(\frac{x}{2} \right)} + 1 \right)} + 2 \tan^{3}{\left(\frac{x}{2} \right)} + 4 \tan^{2}{\left(\frac{x}{2} \right)} - 2 \tan{\left(\frac{x}{2} \right)}}{4 \left(\tan^{4}{\left(\frac{x}{2} \right)} + 2 \tan^{2}{\left(\frac{x}{2} \right)} + 1\right)}$$
(-log(1 + tan(x/2)^2) - 2*tan(x/2) + 2*tan(x/2)^3 + 4*tan(x/2)^2 + tan(x/2)^4*log(1 - tan(x/2)^2 + 2*tan(x/2)) - tan(x/2)^4*log(1 + tan(x/2)^2) - 2*tan(x/2)^2*log(1 + tan(x/2)^2) + 2*tan(x/2)^2*log(1 - tan(x/2)^2 + 2*tan(x/2)) + log(1 - tan(x/2)^2 + 2*tan(x/2)))/(4*(1 + tan(x/2)^4 + 2*tan(x/2)^2))
Potencias
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/ 2 \ / 2\ / 2 \ / 2\ / 2 \ / 2\
| / I*x -I*x \ / I*x -I*x \| | / I*x -I*x \ | 4 | / I*x -I*x \ / I*x -I*x \| 4 | / I*x -I*x \ | 2 | / I*x -I*x \ / I*x -I*x \| 2 | / I*x -I*x \ |
| | --- -----| | --- -----|| | | --- -----| | / I*x -I*x \ | | --- -----| | --- -----|| / I*x -I*x \ | | --- -----| | / I*x -I*x \ | | --- -----| | --- -----|| / I*x -I*x \ | | --- -----| |
| | 2 2 | | 2 2 || | | 2 2 | | | --- -----| | | 2 2 | | 2 2 || | --- -----| | | 2 2 | | | --- -----| | | 2 2 | | 2 2 || | --- -----| | | 2 2 | |
| \- e + e / 2*I*\- e + e /| | \- e + e / | | 2 2 | | \- e + e / 2*I*\- e + e /| | 2 2 | | \- e + e / | | 2 2 | | \- e + e / 2*I*\- e + e /| | 2 2 | | \- e + e / |
log|1 + ------------------ + ---------------------| log|1 - ------------------| \- e + e / *log|1 + ------------------ + ---------------------| \- e + e / *log|1 - ------------------| 2*\- e + e / *log|1 + ------------------ + ---------------------| 2*\- e + e / *log|1 - ------------------|
| 2 I*x -I*x | | 2 | 2 | 2 I*x -I*x | | 2 | 3 | 2 I*x -I*x | | 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 \ | / I*x -I*x \ --- ----- | | / I*x -I*x \ |
| | --- -----| 2 2 | | | --- -----| | | --- -----| | | --- -----| 2 2 | | | --- -----| | | --- -----| | --- -----| | | --- -----| 2 2 | | | --- -----| |
| | 2 2 | e + e | | | 2 2 | | | 2 2 | | | 2 2 | e + e | | | 2 2 | | | 2 2 | | 2 2 | | | 2 2 | e + e | | | 2 2 | |
\ \e + e / / \ \e + e / / 4*\- e + e / \ \e + e / / \ \e + e / / 2*I*\- e + e / 2*I*\- e + e / \ \e + e / / \ \e + e / /
--------------------------------------------------- - ----------------------------------------------- - ------------------------------------------------------------------ + ---------------------------------------------------------------------- - ------------------------------------------------------------------ - ------------------------------------------------------------------ - ----------------------------------------------------------------- - ------------------------------------------------------------------------ + ------------------------------------------------------------------
2 4 2 4 / 2 4\ / 2 4\ / 2 4\ / 2 4\ / 2 4\ / 2 4\ / 2 4\
/ I*x -I*x \ / I*x -I*x \ / I*x -I*x \ / I*x -I*x \ 2 | / I*x -I*x \ / I*x -I*x \ | 4 | / I*x -I*x \ / I*x -I*x \ | 4 | / I*x -I*x \ / I*x -I*x \ | 3 | / I*x -I*x \ / I*x -I*x \ | | / I*x -I*x \ / I*x -I*x \ | 2 | / I*x -I*x \ / I*x -I*x \ | 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 \ | | --- -----| | --- -----| | / I*x -I*x \ | | --- -----| | --- -----| | / I*x -I*x \ | | --- -----| | --- -----| |
| 2 2 | | 2 2 | | 2 2 | | 2 2 | | --- -----| | | 2 2 | | 2 2 | | | --- -----| | | 2 2 | | 2 2 | | | --- -----| | | 2 2 | | 2 2 | | | --- -----| | | 2 2 | | 2 2 | | | --- -----| | | 2 2 | | 2 2 | | | --- -----| | | 2 2 | | 2 2 | | | --- -----| | | 2 2 | | 2 2 | |
8*\- e + e / 4*\- e + e / 8*\- e + e / 4*\- e + e / | 2 2 | | 8*\- e + e / 4*\- e + e / | | 2 2 | | 8*\- e + e / 4*\- e + e / | | 2 2 | | 8*\- e + e / 4*\- e + e / | | 2 2 | | 8*\- e + e / 4*\- e + e / | | 2 2 | | 8*\- e + e / 4*\- e + e / | | 2 2 | | 8*\- e + e / 4*\- e + e / | | 2 2 | | 8*\- e + e / 4*\- e + e / |
4 - -------------------- + -------------------- 4 - -------------------- + -------------------- \e + e / *|4 - -------------------- + --------------------| \e + e / *|4 - -------------------- + --------------------| \e + e / *|4 - -------------------- + --------------------| \e + e / *|4 - -------------------- + --------------------| \e + e /*|4 - -------------------- + --------------------| \e + e / *|4 - -------------------- + --------------------| \e + e / *|4 - -------------------- + --------------------|
2 4 2 4 | 2 4 | | 2 4 | | 2 4 | | 2 4 | | 2 4 | | 2 4 | | 2 4 |
/ 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 \ / 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 \ | | / I*x -I*x \ / I*x -I*x \ | | / I*x -I*x \ / I*x -I*x \ |
| --- -----| | --- -----| | --- -----| | --- -----| | | --- -----| | --- -----| | | | --- -----| | --- -----| | | | --- -----| | --- -----| | | | --- -----| | --- -----| | | | --- -----| | --- -----| | | | --- -----| | --- -----| | | | --- -----| | --- -----| |
| 2 2 | | 2 2 | | 2 2 | | 2 2 | | | 2 2 | | 2 2 | | | | 2 2 | | 2 2 | | | | 2 2 | | 2 2 | | | | 2 2 | | 2 2 | | | | 2 2 | | 2 2 | | | | 2 2 | | 2 2 | | | | 2 2 | | 2 2 | |
\e + e / \e + e / \e + e / \e + e / \ \e + e / \e + e / / \ \e + e / \e + e / / \ \e + e / \e + e / / \ \e + e / \e + e / / \ \e + e / \e + e / / \ \e + e / \e + e / / \ \e + e / \e + e / /
$$- \frac{\left(- e^{\frac{i x}{2}} + e^{- \frac{i x}{2}}\right)^{4} \log{\left(- \frac{\left(- e^{\frac{i x}{2}} + e^{- \frac{i x}{2}}\right)^{2}}{\left(e^{\frac{i x}{2}} + e^{- \frac{i x}{2}}\right)^{2}} + 1 \right)}}{\left(e^{\frac{i x}{2}} + e^{- \frac{i x}{2}}\right)^{4} \left(\frac{4 \left(- e^{\frac{i x}{2}} + e^{- \frac{i x}{2}}\right)^{4}}{\left(e^{\frac{i x}{2}} + e^{- \frac{i x}{2}}\right)^{4}} - \frac{8 \left(- e^{\frac{i x}{2}} + e^{- \frac{i x}{2}}\right)^{2}}{\left(e^{\frac{i x}{2}} + e^{- \frac{i x}{2}}\right)^{2}} + 4\right)} + \frac{\left(- e^{\frac{i x}{2}} + e^{- \frac{i x}{2}}\right)^{4} \log{\left(\frac{\left(- e^{\frac{i x}{2}} + e^{- \frac{i x}{2}}\right)^{2}}{\left(e^{\frac{i x}{2}} + e^{- \frac{i x}{2}}\right)^{2}} + \frac{2 i \left(- e^{\frac{i x}{2}} + e^{- \frac{i x}{2}}\right)}{e^{\frac{i x}{2}} + e^{- \frac{i x}{2}}} + 1 \right)}}{\left(e^{\frac{i x}{2}} + e^{- \frac{i x}{2}}\right)^{4} \left(\frac{4 \left(- e^{\frac{i x}{2}} + e^{- \frac{i x}{2}}\right)^{4}}{\left(e^{\frac{i x}{2}} + e^{- \frac{i x}{2}}\right)^{4}} - \frac{8 \left(- e^{\frac{i x}{2}} + e^{- \frac{i x}{2}}\right)^{2}}{\left(e^{\frac{i x}{2}} + e^{- \frac{i x}{2}}\right)^{2}} + 4\right)} - \frac{2 i \left(- e^{\frac{i x}{2}} + e^{- \frac{i x}{2}}\right)^{3}}{\left(e^{\frac{i x}{2}} + e^{- \frac{i x}{2}}\right)^{3} \left(\frac{4 \left(- e^{\frac{i x}{2}} + e^{- \frac{i x}{2}}\right)^{4}}{\left(e^{\frac{i x}{2}} + e^{- \frac{i x}{2}}\right)^{4}} - \frac{8 \left(- e^{\frac{i x}{2}} + e^{- \frac{i x}{2}}\right)^{2}}{\left(e^{\frac{i x}{2}} + e^{- \frac{i x}{2}}\right)^{2}} + 4\right)} + \frac{2 \left(- e^{\frac{i x}{2}} + e^{- \frac{i x}{2}}\right)^{2} \log{\left(- \frac{\left(- e^{\frac{i x}{2}} + e^{- \frac{i x}{2}}\right)^{2}}{\left(e^{\frac{i x}{2}} + e^{- \frac{i x}{2}}\right)^{2}} + 1 \right)}}{\left(e^{\frac{i x}{2}} + e^{- \frac{i x}{2}}\right)^{2} \left(\frac{4 \left(- e^{\frac{i x}{2}} + e^{- \frac{i x}{2}}\right)^{4}}{\left(e^{\frac{i x}{2}} + e^{- \frac{i x}{2}}\right)^{4}} - \frac{8 \left(- e^{\frac{i x}{2}} + e^{- \frac{i x}{2}}\right)^{2}}{\left(e^{\frac{i x}{2}} + e^{- \frac{i x}{2}}\right)^{2}} + 4\right)} - \frac{2 \left(- e^{\frac{i x}{2}} + e^{- \frac{i x}{2}}\right)^{2} \log{\left(\frac{\left(- e^{\frac{i x}{2}} + e^{- \frac{i x}{2}}\right)^{2}}{\left(e^{\frac{i x}{2}} + e^{- \frac{i x}{2}}\right)^{2}} + \frac{2 i \left(- e^{\frac{i x}{2}} + e^{- \frac{i x}{2}}\right)}{e^{\frac{i x}{2}} + e^{- \frac{i x}{2}}} + 1 \right)}}{\left(e^{\frac{i x}{2}} + e^{- \frac{i x}{2}}\right)^{2} \left(\frac{4 \left(- e^{\frac{i x}{2}} + e^{- \frac{i x}{2}}\right)^{4}}{\left(e^{\frac{i x}{2}} + e^{- \frac{i x}{2}}\right)^{4}} - \frac{8 \left(- e^{\frac{i x}{2}} + e^{- \frac{i x}{2}}\right)^{2}}{\left(e^{\frac{i x}{2}} + e^{- \frac{i x}{2}}\right)^{2}} + 4\right)} - \frac{4 \left(- e^{\frac{i x}{2}} + e^{- \frac{i x}{2}}\right)^{2}}{\left(e^{\frac{i x}{2}} + e^{- \frac{i x}{2}}\right)^{2} \left(\frac{4 \left(- e^{\frac{i x}{2}} + e^{- \frac{i x}{2}}\right)^{4}}{\left(e^{\frac{i x}{2}} + e^{- \frac{i x}{2}}\right)^{4}} - \frac{8 \left(- e^{\frac{i x}{2}} + e^{- \frac{i x}{2}}\right)^{2}}{\left(e^{\frac{i x}{2}} + e^{- \frac{i x}{2}}\right)^{2}} + 4\right)} - \frac{2 i \left(- e^{\frac{i x}{2}} + e^{- \frac{i x}{2}}\right)}{\left(e^{\frac{i x}{2}} + e^{- \frac{i x}{2}}\right) \left(\frac{4 \left(- e^{\frac{i x}{2}} + e^{- \frac{i x}{2}}\right)^{4}}{\left(e^{\frac{i x}{2}} + e^{- \frac{i x}{2}}\right)^{4}} - \frac{8 \left(- e^{\frac{i x}{2}} + e^{- \frac{i x}{2}}\right)^{2}}{\left(e^{\frac{i x}{2}} + e^{- \frac{i x}{2}}\right)^{2}} + 4\right)} - \frac{\log{\left(- \frac{\left(- e^{\frac{i x}{2}} + e^{- \frac{i x}{2}}\right)^{2}}{\left(e^{\frac{i x}{2}} + e^{- \frac{i x}{2}}\right)^{2}} + 1 \right)}}{\frac{4 \left(- e^{\frac{i x}{2}} + e^{- \frac{i x}{2}}\right)^{4}}{\left(e^{\frac{i x}{2}} + e^{- \frac{i x}{2}}\right)^{4}} - \frac{8 \left(- e^{\frac{i x}{2}} + e^{- \frac{i x}{2}}\right)^{2}}{\left(e^{\frac{i x}{2}} + e^{- \frac{i x}{2}}\right)^{2}} + 4} + \frac{\log{\left(\frac{\left(- e^{\frac{i x}{2}} + e^{- \frac{i x}{2}}\right)^{2}}{\left(e^{\frac{i x}{2}} + e^{- \frac{i x}{2}}\right)^{2}} + \frac{2 i \left(- e^{\frac{i x}{2}} + e^{- \frac{i x}{2}}\right)}{e^{\frac{i x}{2}} + e^{- \frac{i x}{2}}} + 1 \right)}}{\frac{4 \left(- e^{\frac{i x}{2}} + e^{- \frac{i x}{2}}\right)^{4}}{\left(e^{\frac{i x}{2}} + e^{- \frac{i x}{2}}\right)^{4}} - \frac{8 \left(- e^{\frac{i x}{2}} + e^{- \frac{i x}{2}}\right)^{2}}{\left(e^{\frac{i x}{2}} + e^{- \frac{i x}{2}}\right)^{2}} + 4}$$
/ 2/x\ /x\\ / 2/x\\ /x\ 3/x\ 2/x\ 4/x\ / 2/x\ /x\\ 4/x\ / 2/x\\ 2/x\ / 2/x\\ 2/x\ / 2/x\ /x\\
log|1 - tan |-| + 2*tan|-|| log|1 + tan |-|| 2*tan|-| 2*tan |-| 4*tan |-| tan |-|*log|1 - tan |-| + 2*tan|-|| tan |-|*log|1 + tan |-|| 2*tan |-|*log|1 + tan |-|| 2*tan |-|*log|1 - tan |-| + 2*tan|-||
\ \2/ \2// \ \2// \2/ \2/ \2/ \2/ \ \2/ \2// \2/ \ \2// \2/ \ \2// \2/ \ \2/ \2//
--------------------------- - ------------------------- - ------------------------- + ------------------------- + ------------------------- + ----------------------------------- - ------------------------- - -------------------------- + -------------------------------------
4/x\ 2/x\ 4/x\ 2/x\ 4/x\ 2/x\ 4/x\ 2/x\ 4/x\ 2/x\ 4/x\ 2/x\ 4/x\ 2/x\ 4/x\ 2/x\ 4/x\ 2/x\
4 + 4*tan |-| + 8*tan |-| 4 + 4*tan |-| + 8*tan |-| 4 + 4*tan |-| + 8*tan |-| 4 + 4*tan |-| + 8*tan |-| 4 + 4*tan |-| + 8*tan |-| 4 + 4*tan |-| + 8*tan |-| 4 + 4*tan |-| + 8*tan |-| 4 + 4*tan |-| + 8*tan |-| 4 + 4*tan |-| + 8*tan |-|
\2/ \2/ \2/ \2/ \2/ \2/ \2/ \2/ \2/ \2/ \2/ \2/ \2/ \2/ \2/ \2/ \2/ \2/
$$- \frac{\log{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1 \right)} \tan^{4}{\left(\frac{x}{2} \right)}}{4 \tan^{4}{\left(\frac{x}{2} \right)} + 8 \tan^{2}{\left(\frac{x}{2} \right)} + 4} - \frac{2 \log{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1 \right)} \tan^{2}{\left(\frac{x}{2} \right)}}{4 \tan^{4}{\left(\frac{x}{2} \right)} + 8 \tan^{2}{\left(\frac{x}{2} \right)} + 4} - \frac{\log{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1 \right)}}{4 \tan^{4}{\left(\frac{x}{2} \right)} + 8 \tan^{2}{\left(\frac{x}{2} \right)} + 4} + \frac{\log{\left(- \tan^{2}{\left(\frac{x}{2} \right)} + 2 \tan{\left(\frac{x}{2} \right)} + 1 \right)} \tan^{4}{\left(\frac{x}{2} \right)}}{4 \tan^{4}{\left(\frac{x}{2} \right)} + 8 \tan^{2}{\left(\frac{x}{2} \right)} + 4} + \frac{2 \log{\left(- \tan^{2}{\left(\frac{x}{2} \right)} + 2 \tan{\left(\frac{x}{2} \right)} + 1 \right)} \tan^{2}{\left(\frac{x}{2} \right)}}{4 \tan^{4}{\left(\frac{x}{2} \right)} + 8 \tan^{2}{\left(\frac{x}{2} \right)} + 4} + \frac{\log{\left(- \tan^{2}{\left(\frac{x}{2} \right)} + 2 \tan{\left(\frac{x}{2} \right)} + 1 \right)}}{4 \tan^{4}{\left(\frac{x}{2} \right)} + 8 \tan^{2}{\left(\frac{x}{2} \right)} + 4} + \frac{2 \tan^{3}{\left(\frac{x}{2} \right)}}{4 \tan^{4}{\left(\frac{x}{2} \right)} + 8 \tan^{2}{\left(\frac{x}{2} \right)} + 4} + \frac{4 \tan^{2}{\left(\frac{x}{2} \right)}}{4 \tan^{4}{\left(\frac{x}{2} \right)} + 8 \tan^{2}{\left(\frac{x}{2} \right)} + 4} - \frac{2 \tan{\left(\frac{x}{2} \right)}}{4 \tan^{4}{\left(\frac{x}{2} \right)} + 8 \tan^{2}{\left(\frac{x}{2} \right)} + 4}$$
log(1 - tan(x/2)^2 + 2*tan(x/2))/(4 + 4*tan(x/2)^4 + 8*tan(x/2)^2) - log(1 + tan(x/2)^2)/(4 + 4*tan(x/2)^4 + 8*tan(x/2)^2) - 2*tan(x/2)/(4 + 4*tan(x/2)^4 + 8*tan(x/2)^2) + 2*tan(x/2)^3/(4 + 4*tan(x/2)^4 + 8*tan(x/2)^2) + 4*tan(x/2)^2/(4 + 4*tan(x/2)^4 + 8*tan(x/2)^2) + tan(x/2)^4*log(1 - tan(x/2)^2 + 2*tan(x/2))/(4 + 4*tan(x/2)^4 + 8*tan(x/2)^2) - tan(x/2)^4*log(1 + tan(x/2)^2)/(4 + 4*tan(x/2)^4 + 8*tan(x/2)^2) - 2*tan(x/2)^2*log(1 + tan(x/2)^2)/(4 + 4*tan(x/2)^4 + 8*tan(x/2)^2) + 2*tan(x/2)^2*log(1 - tan(x/2)^2 + 2*tan(x/2))/(4 + 4*tan(x/2)^4 + 8*tan(x/2)^2)
Combinatoria
[src]
/ 2/x\\ /x\ 3/x\ 2/x\ 4/x\ / 2/x\ /x\\ 4/x\ / 2/x\\ 2/x\ / 2/x\\ 2/x\ / 2/x\ /x\\ / 2/x\ /x\\
- log|1 + tan |-|| - 2*tan|-| + 2*tan |-| + 4*tan |-| + tan |-|*log|1 - tan |-| + 2*tan|-|| - tan |-|*log|1 + tan |-|| - 2*tan |-|*log|1 + tan |-|| + 2*tan |-|*log|1 - tan |-| + 2*tan|-|| + log|1 - tan |-| + 2*tan|-||
\ \2// \2/ \2/ \2/ \2/ \ \2/ \2// \2/ \ \2// \2/ \ \2// \2/ \ \2/ \2// \ \2/ \2//
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
2
/ 2/x\\
4*|1 + tan |-||
\ \2//
$$\frac{- \log{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1 \right)} \tan^{4}{\left(\frac{x}{2} \right)} - 2 \log{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1 \right)} \tan^{2}{\left(\frac{x}{2} \right)} - \log{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1 \right)} + \log{\left(- \tan^{2}{\left(\frac{x}{2} \right)} + 2 \tan{\left(\frac{x}{2} \right)} + 1 \right)} \tan^{4}{\left(\frac{x}{2} \right)} + 2 \log{\left(- \tan^{2}{\left(\frac{x}{2} \right)} + 2 \tan{\left(\frac{x}{2} \right)} + 1 \right)} \tan^{2}{\left(\frac{x}{2} \right)} + \log{\left(- \tan^{2}{\left(\frac{x}{2} \right)} + 2 \tan{\left(\frac{x}{2} \right)} + 1 \right)} + 2 \tan^{3}{\left(\frac{x}{2} \right)} + 4 \tan^{2}{\left(\frac{x}{2} \right)} - 2 \tan{\left(\frac{x}{2} \right)}}{4 \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}}$$
(-log(1 + tan(x/2)^2) - 2*tan(x/2) + 2*tan(x/2)^3 + 4*tan(x/2)^2 + tan(x/2)^4*log(1 - tan(x/2)^2 + 2*tan(x/2)) - tan(x/2)^4*log(1 + tan(x/2)^2) - 2*tan(x/2)^2*log(1 + tan(x/2)^2) + 2*tan(x/2)^2*log(1 - tan(x/2)^2 + 2*tan(x/2)) + log(1 - tan(x/2)^2 + 2*tan(x/2)))/(4*(1 + tan(x/2)^2)^2)
Compilar la expresión
[src]
/ 2/x\ /x\\ / 2/x\\ /x\ 3/x\ 2/x\ 4/x\ / 2/x\ /x\\ 4/x\ / 2/x\\ 2/x\ / 2/x\\ 2/x\ / 2/x\ /x\\
log|1 - tan |-| + 2*tan|-|| log|1 + tan |-|| 2*tan|-| 2*tan |-| 4*tan |-| tan |-|*log|1 - tan |-| + 2*tan|-|| tan |-|*log|1 + tan |-|| 2*tan |-|*log|1 + tan |-|| 2*tan |-|*log|1 - tan |-| + 2*tan|-||
\ \2/ \2// \ \2// \2/ \2/ \2/ \2/ \ \2/ \2// \2/ \ \2// \2/ \ \2// \2/ \ \2/ \2//
--------------------------- - ------------------------- - ------------------------- + ------------------------- + ------------------------- + ----------------------------------- - ------------------------- - -------------------------- + -------------------------------------
4/x\ 2/x\ 4/x\ 2/x\ 4/x\ 2/x\ 4/x\ 2/x\ 4/x\ 2/x\ 4/x\ 2/x\ 4/x\ 2/x\ 4/x\ 2/x\ 4/x\ 2/x\
4 + 4*tan |-| + 8*tan |-| 4 + 4*tan |-| + 8*tan |-| 4 + 4*tan |-| + 8*tan |-| 4 + 4*tan |-| + 8*tan |-| 4 + 4*tan |-| + 8*tan |-| 4 + 4*tan |-| + 8*tan |-| 4 + 4*tan |-| + 8*tan |-| 4 + 4*tan |-| + 8*tan |-| 4 + 4*tan |-| + 8*tan |-|
\2/ \2/ \2/ \2/ \2/ \2/ \2/ \2/ \2/ \2/ \2/ \2/ \2/ \2/ \2/ \2/ \2/ \2/
$$\frac{\log{\left(\left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right) + 2 \tan{\left(\frac{x}{2} \right)} \right)} \tan^{4}{\left(\frac{x}{2} \right)}}{4 \tan^{4}{\left(\frac{x}{2} \right)} + 8 \tan^{2}{\left(\frac{x}{2} \right)} + 4} + \frac{2 \log{\left(\left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right) + 2 \tan{\left(\frac{x}{2} \right)} \right)} \tan^{2}{\left(\frac{x}{2} \right)}}{4 \tan^{4}{\left(\frac{x}{2} \right)} + 8 \tan^{2}{\left(\frac{x}{2} \right)} + 4} + \frac{\log{\left(\left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right) + 2 \tan{\left(\frac{x}{2} \right)} \right)}}{4 \tan^{4}{\left(\frac{x}{2} \right)} + 8 \tan^{2}{\left(\frac{x}{2} \right)} + 4} - \frac{\log{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1 \right)} \tan^{4}{\left(\frac{x}{2} \right)}}{4 \tan^{4}{\left(\frac{x}{2} \right)} + 8 \tan^{2}{\left(\frac{x}{2} \right)} + 4} - \frac{2 \log{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1 \right)} \tan^{2}{\left(\frac{x}{2} \right)}}{4 \tan^{4}{\left(\frac{x}{2} \right)} + 8 \tan^{2}{\left(\frac{x}{2} \right)} + 4} - \frac{\log{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1 \right)}}{4 \tan^{4}{\left(\frac{x}{2} \right)} + 8 \tan^{2}{\left(\frac{x}{2} \right)} + 4} + \frac{2 \tan^{3}{\left(\frac{x}{2} \right)}}{4 \tan^{4}{\left(\frac{x}{2} \right)} + 8 \tan^{2}{\left(\frac{x}{2} \right)} + 4} + \frac{4 \tan^{2}{\left(\frac{x}{2} \right)}}{4 \tan^{4}{\left(\frac{x}{2} \right)} + 8 \tan^{2}{\left(\frac{x}{2} \right)} + 4} - \frac{2 \tan{\left(\frac{x}{2} \right)}}{4 \tan^{4}{\left(\frac{x}{2} \right)} + 8 \tan^{2}{\left(\frac{x}{2} \right)} + 4}$$
log(1 - tan(x/2)^2 + 2*tan(x/2))/(4 + 4*tan(x/2)^4 + 8*tan(x/2)^2) - log(1 + tan(x/2)^2)/(4 + 4*tan(x/2)^4 + 8*tan(x/2)^2) - 2*tan(x/2)/(4 + 4*tan(x/2)^4 + 8*tan(x/2)^2) + 2*tan(x/2)^3/(4 + 4*tan(x/2)^4 + 8*tan(x/2)^2) + 4*tan(x/2)^2/(4 + 4*tan(x/2)^4 + 8*tan(x/2)^2) + tan(x/2)^4*log(1 - tan(x/2)^2 + 2*tan(x/2))/(4 + 4*tan(x/2)^4 + 8*tan(x/2)^2) - tan(x/2)^4*log(1 + tan(x/2)^2)/(4 + 4*tan(x/2)^4 + 8*tan(x/2)^2) - 2*tan(x/2)^2*log(1 + tan(x/2)^2)/(4 + 4*tan(x/2)^4 + 8*tan(x/2)^2) + 2*tan(x/2)^2*log(1 - tan(x/2)^2 + 2*tan(x/2))/(4 + 4*tan(x/2)^4 + 8*tan(x/2)^2)