Integral of 1/(x^n+1) from 0 to ∞

\begin{aligned} I = \int_0^\infty \dfrac{1}{(x)^n+1} dx  = \dfrac{\pi}{n\sin{( \frac{\pi}{n}} )} \end{aligned}

\begin{aligned} \therefore \rho=\left[ (x)^n+1 \right]^{-1} \:;\: \rho \mid_0^1 \: \Longrightarrow d\rho=\dfrac{-n(x)^{n-1}}{\left[ (x)^n+1 \right]^2} \: dx
\end{aligned}

\begin{aligned} \therefore
(n)^{-1} \int_0^1 \left[ (1-\rho)^{(n)^{-1}-1}(\rho)^{-(n)^{-1}} \right] d\rho =
\dfrac{\beta (\frac{1}{n},1-\frac{1}{n})}{n}
\end{aligned}

\begin{aligned}
\dfrac{\beta (\frac{1}{n},1-\frac{1}{n})}{n} =
\dfrac {\Gamma (\frac{1}{n}) \Gamma (1-\frac{1}{n})}{n} =
\dfrac{\pi}{n\sin{( \frac{\pi}{n}} )}
\end{aligned}

References:

Beta Function
https://en.wikipedia.org/wiki/Beta_function
Gamma Function
https://en.wikipedia.org/wiki/Gamma_function
Euler Reflection Formula
https://proofwiki.org/wiki/Euler's_Reflection_Formula
Euler Sine Formula
https://proofwiki.org/wiki/Euler_Formula_for_Sine_Function


Dr. Peyam
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