Fractional integral with two parameters

Question

How can i find the exact value of $$\int\limits_0^\infty {{{{x^m}} \over {1 + {x^p}}}dx} $$ with $p < m + 1$. Thank you.

Answer 1

Through the substitution $\frac{1}{1+x^p}=u$ the given integral boils down to a value of Euler's Beta function, namely $$ \int_{0}^{\infty}\frac{x^m}{1+x^p}\,dx = \frac{\pi}{p\sin\left(\frac{\pi(m+1)}{p}\right)} $$ due to the reflection formula for the $\Gamma$ function.
The above identity holds as soon as $\text{Re}(m)>-1$ and $\text{Re}(m-p)<-1$.