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