Question

RETO.
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Calcular la siguiente integral

$\displaystyle\\\int_{0}^{+\infty}\dfrac{1}{x^n+1}~dx$

Answer 1

$\displaystyle\\I=\int_0^{+\infty}\dfrac{dx}{x^n+1}\\ \\\text{1. Cambio de variable: }t=\dfrac{1}{x^n+1}\to x=\left(\dfrac{1}{t}-1\right)^{1/n}\\ \\dx=-\dfrac{1}{nt^2}\left(\dfrac{1}{t}-1\right)^{\frac{1-n}{n}}dt$

$\text{2. Sustituci\'on: }\\\displaystyle\\I=\int_1^0t\cdot \left(-\dfrac{1}{nt^2}\right)\left(\dfrac{1}{t}-1\right)^{\frac{1-n}{n}}~dt\\ \\ \\I=\dfrac{1}{n}\int_0^1t^{-1/n}(1-t)^{\frac{1}{n}-1}~dt\\ \\\\I=\dfrac{1}{n} \text{B}\left(1-\dfrac{1}{n},\dfrac{1}{n}\right) \\ \\ \\I=\dfrac{\Gamma\left(1-\dfrac{1}{n}\right)\Gamma\left(\dfrac{1}{n}\right)}{n\Gamma(1)}\\ \\ \\I=\dfrac{1}{n}\Gamma\left(1-\dfrac{1}{n}\right)\Gamma\left(\dfrac{1}{n}\right)$

$\text{F\'ormula de reflexi\'on de Euler}\\ \\\\I=\dfrac{\pi}{n}\csc\dfrac{\pi}{n}$