# Apostol equation for partial fraction simplification

In Apostol, the case that the partial fraction is presented as $$\frac{C}{(u^2 + a^2)^m}$$ can be done with the following reduction formula:

$$\int \frac{du}{(u^2 + a^2)^m} = \frac{1}{2a^2(m-1)} \frac{u}{(u^2 + a^2)^{m-1}} + \frac{2m - 3}{2a^2(m-1)} \int \frac{du}{(u^2 + a^2)^{m-1}}$$

The author states that this can be shown using integration by parts. I cannot figure how - i tried taking $$dv = (u^2 + a^2)^{m-1}$$ or $$dv = 1 \Rightarrow v = u$$. In case the $$m = 2$$, the integral can be done with trigonometric substitution. However, this general case is not as clear. Can someone show how this is obtained with integration by parts?

I suggest that you write $$\frac{1}{(u^2+a^2)^{m-1}} = u\cdot\frac{u}{(u^2+a^2)^m}+\frac{a^2}{(u^2+a^2)^m},$$ and integrate by parts in the first term in the right-hand side, and then rearrange your terms.