Integration: arctan(x) rule derivation

**Greg98** · November 3rd 2010, 12:03 PM

Hello,
the task is to derive integration rule (2) from rule (1).

(1) $\int \frac{1}{x^2+1} dx = \arctan (x)$

(2) $\int \frac{1}{x^2+c^2} dx = \frac{1}{c} \arctan (\frac{x}{c})$

$c$ is constant and $>0$

I tried to massage rule (1) with basic operations, but I always ended up with too many x's and c's. Also, I couldn't find any help using forum's search function. So, any help is welcome. Mucho thanks in advance!

**TheEmptySet** · November 3rd 2010, 12:11 PM

let $cu=x \implies cdu = dx$

so your integral becomes

$\displaystyle \int \frac{dx}{x^2+c}=\int \frac{c du }{c^2u^2+c^2}=\frac{1}{c}\int \frac{du}{u^2+1}$

and the rest follows.

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