calculus - Is there a general formula for $I(m,n)$?

Consider the integral

$$I(m,n):=\int_0^{\infty} \frac{x^m}{x^n+1}\,\mathrm dx$$

For $m=0$, a general formula is $$I(0,n)=\frac{\frac{\pi}{n}}{\sin\left(\frac{\pi}{n}\right)}$$

Some other values are $$I(1,3)=\frac{2\pi}{3\sqrt{3}}$$ $$I(1,4)=\frac{\pi}{4}$$ $$I(2,4)=\frac{\pi}{2\sqrt{2}}$$

For natural $m,n$ the integral exists if and only if $n\ge m+2$.

Is there a general formula for $I(m,n)$ with integers $m,n$ and $0\le m\le n-2$ ?

Answer

We can use contour integration to arrive at the general result. Note that

$$\begin{align} \oint_C \frac{z^m}{z^n+1}\,dz&=2\pi i \text{Res}\left(\frac{z^m}{z^n+1}, z=e^{i\pi/n}\right)\\\\ &=-2\pi i \frac{e^{i\pi(m+1)/n}}{n}\tag 1 \end{align}$$

where $C$ is the "pie slice" contour comprised of (i) the real-line segment from $0$ to $R$, where $R>1$, (ii) the circular arc of radius $R$ that begins at $R$ and ends at $Re^{i2\pi/n}$, and $(3)$ the straight line segment from $Re^{i2\pi/n}$ to $0$.

Then, we can write

$$\oint_C \frac{z^m}{z^n+1}\,dz=\int_0^R \frac{x^m}{x^n+1}\,dx+\int_0^{2\pi/2}\frac{R^me^{im\phi}}{R^ne^{in\phi}+1}\,iRe^{i\phi}\,d\phi-\int_0^R \frac{x^me^{i2\pi m/n}}{x^n+1}e^{i2\pi/n}\,dx \tag 2$$

If $n>m+1$, then as $R\to \infty$, the second integral on the right-hand side of $(2)$ vanishes and we find that

$$\bbox[5px,border:2px solid #C0A000]{\int_0^\infty \frac{x^m}{x^n+1}\,dx=2\pi i\frac{e^{i\pi(m+1)/n}}{n(e^{i2\pi(m+1)/n}-1)}=\frac{\pi/n}{\sin(\pi(m+1)/n)}}$$