Integral of Arctangent Function

Theorem

 * $\displaystyle \int \arctan x \ \mathrm d x = x \arctan x - \frac 1 2 \ln \left({x^2 + 1}\right) + C$

Proof

 * $\displaystyle \int \arctan x \ \mathrm d x = \int 1 \cdot \arctan x \ \mathrm d x$

Using: we obtain:
 * Integration by Parts
 * Derivative of Arctangent Function
 * Integration of a Constant


 * $\displaystyle \int \arctan x \ \mathrm d x = x \arctan x - \int \dfrac x {1 + x^2} \ \mathrm dx$

Substitute:

Then: