Sr Examen
Integral de log((1-x^n)/(1+x^n)) dx
Solución
Ha introducido
[src]
n / | | / n\ | |1 - x | | log|------| dx | | n| | \1 + x / | / 0
$$\int\limits_{0}^{n} \log{\left(\frac{1 - x^{n}}{x^{n} + 1} \right)}\, dx$$
Integral(log((1 - x^n)/(1 + x^n)), (x, 0, n))
Respuesta (Indefinida)
[src]
/ | / n /1 1 \ / 2*n 1 1 \ n /1 1 \ / 2*n 1 1 \\ | / n\ / n\ | x*x *Gamma|- + ---|*lerchphi|x , 1, - + ---| x*x *Gamma|- + ---|*lerchphi|x , 1, - + ---|| | |1 - x | |1 - x | | \2 2*n/ \ 2 2*n/ \2 2*n/ \ 2 2*n/| | log|------| dx = C + x*log|------| - 2*n*|- ---------------------------------------------- - ----------------------------------------------| | | n| | n| | /3 1 \ 2 /3 1 \ | | \1 + x / \1 + x / | 4*n*Gamma|- + ---| 4*n *Gamma|- + ---| | | \ \2 2*n/ \2 2*n/ / /
$$\int \log{\left(\frac{1 - x^{n}}{x^{n} + 1} \right)}\, dx = C - 2 n \left(- \frac{x x^{n} \Phi\left(x^{2 n}, 1, \frac{1}{2} + \frac{1}{2 n}\right) \Gamma\left(\frac{1}{2} + \frac{1}{2 n}\right)}{4 n \Gamma\left(\frac{3}{2} + \frac{1}{2 n}\right)} - \frac{x x^{n} \Phi\left(x^{2 n}, 1, \frac{1}{2} + \frac{1}{2 n}\right) \Gamma\left(\frac{1}{2} + \frac{1}{2 n}\right)}{4 n^{2} \Gamma\left(\frac{3}{2} + \frac{1}{2 n}\right)}\right) + x \log{\left(\frac{1 - x^{n}}{x^{n} + 1} \right)}$$
Respuesta
[src]
/ n /1 1 \ / 2*n 1 1 \ n /1 1 \ / 2*n 1 1 \\
/ n\ | n *Gamma|- + ---|*lerchphi|n , 1, - + ---| n *Gamma|- + ---|*lerchphi|n , 1, - + ---||
|1 - n | | \2 2*n/ \ 2 2*n/ \2 2*n/ \ 2 2*n/|
n*log|------| - 2*n*|- -------------------------------------------- - --------------------------------------------|
| n| | /3 1 \ /3 1 \ |
\1 + n / | 4*Gamma|- + ---| 4*n*Gamma|- + ---| |
\ \2 2*n/ \2 2*n/ /
$$- 2 n \left(- \frac{n^{n} \Phi\left(n^{2 n}, 1, \frac{1}{2} + \frac{1}{2 n}\right) \Gamma\left(\frac{1}{2} + \frac{1}{2 n}\right)}{4 \Gamma\left(\frac{3}{2} + \frac{1}{2 n}\right)} - \frac{n^{n} \Phi\left(n^{2 n}, 1, \frac{1}{2} + \frac{1}{2 n}\right) \Gamma\left(\frac{1}{2} + \frac{1}{2 n}\right)}{4 n \Gamma\left(\frac{3}{2} + \frac{1}{2 n}\right)}\right) + n \log{\left(\frac{1 - n^{n}}{n^{n} + 1} \right)}$$
=
=
/ n /1 1 \ / 2*n 1 1 \ n /1 1 \ / 2*n 1 1 \\
/ n\ | n *Gamma|- + ---|*lerchphi|n , 1, - + ---| n *Gamma|- + ---|*lerchphi|n , 1, - + ---||
|1 - n | | \2 2*n/ \ 2 2*n/ \2 2*n/ \ 2 2*n/|
n*log|------| - 2*n*|- -------------------------------------------- - --------------------------------------------|
| n| | /3 1 \ /3 1 \ |
\1 + n / | 4*Gamma|- + ---| 4*n*Gamma|- + ---| |
\ \2 2*n/ \2 2*n/ /
$$- 2 n \left(- \frac{n^{n} \Phi\left(n^{2 n}, 1, \frac{1}{2} + \frac{1}{2 n}\right) \Gamma\left(\frac{1}{2} + \frac{1}{2 n}\right)}{4 \Gamma\left(\frac{3}{2} + \frac{1}{2 n}\right)} - \frac{n^{n} \Phi\left(n^{2 n}, 1, \frac{1}{2} + \frac{1}{2 n}\right) \Gamma\left(\frac{1}{2} + \frac{1}{2 n}\right)}{4 n \Gamma\left(\frac{3}{2} + \frac{1}{2 n}\right)}\right) + n \log{\left(\frac{1 - n^{n}}{n^{n} + 1} \right)}$$
n*log((1 - n^n)/(1 + n^n)) - 2*n*(-n^n*gamma(1/2 + 1/(2*n))*lerchphi(n^(2*n), 1, 1/2 + 1/(2*n))/(4*gamma(3/2 + 1/(2*n))) - n^n*gamma(1/2 + 1/(2*n))*lerchphi(n^(2*n), 1, 1/2 + 1/(2*n))/(4*n*gamma(3/2 + 1/(2*n))))
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.