# 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?

## 1 Answer

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.

• Cool, thank you! Got it! @mickep, how did you come up with this transformation? Did you know it beforehand, solved similar examples, or you just see the pattern somehow? I think I myself would have never found how to integrate this, but with your transformation it is so simple! – John Aug 13 at 20:17
• @John Glad it helped. I've seen and done similar manipulations before. – mickep Aug 14 at 5:02