Definite Integral to Infinity of Reciprocal of 1 plus Power of x/Corollary

Theorem

$\ds \int_0^\infty \frac 1 {a^n + x^n} \rd x = \frac \pi {n a^{n - 1} } \map \csc {\frac \pi n}$

where:

$n$ is a real number greater than 1

$\csc$ is the cosecant function

$a \ne 0$.

Proof

\(\ds \int_0^\infty \frac 1 {a^n + x^n} \rd x\)

\(=\)

\(\ds \frac 1 {a^n} \int_0^\infty \frac 1 {1 + \paren {\frac x a}^n} \rd x\)

\(\ds \)

\(=\)

\(\ds \frac 1 {a^n} \cdot \frac 1 {\frac 1 a} \int_0^\infty \frac 1 {1 + \paren {\frac x a}^n} \map \rd {\frac x a}\)

Primitive of Function of Constant Multiple

\(\ds \)

\(=\)

\(\ds \frac a {a^n} \cdot \frac \pi n \map \csc {\frac \pi n}\)

Definite Integral to Infinity of $\dfrac 1 {1 + x^n}$

\(\ds \)

\(=\)

\(\ds \frac \pi {n a^{n - 1} } \map \csc {\frac \pi n}\)

$\blacksquare$