My textbook teaches that any simple fraction of two polynomials can be divided into factors and then integrated. One of those possible factors is

$\displaystyle \left(\frac{1}{t^2 + b^2}\right)^\alpha$ where t is the argument and b is a constant.

It then goes on to say that in order to integrate that fraction, one must apply the formula $\displaystyle I=\frac{t}{2b^2{(\alpha-2)}(t^2+b^2)^{\alpha-1}} + \frac{2\alpha-3}{2b^2(\alpha-1)}I_{\alpha-1}$

Where $\displaystyle I$ is the integral of the fraction and $\displaystyle I_{\alpha-1}$ is a new integral with the power of the denominator reduced by one.

Can somebody explain to me where that formula comes from.

My first thought was that it looked like integration by parts, but that didn't get me anywhere.