# Primitive of Arccotangent of x over a over x

## Theorem

$\displaystyle \int \frac {\operatorname{arccot} \frac x a \ \mathrm d x} x = \frac \pi 2 \ln \left\vert{x}\right\vert - \int \frac {\arctan \frac x a \ \mathrm d x} x$

## Proof

 $\displaystyle \int \frac {\operatorname{arccot} \frac x a \ \mathrm d x} x$ $=$ $\displaystyle \int \frac {\left({\frac \pi 2 - \arctan \frac x a}\right) \ \mathrm d x} x$ Sum of Arctangent and Arccotangent $\displaystyle$ $=$ $\displaystyle \frac \pi 2 \int \frac {\mathrm d x} x - \int \frac {\arctan \frac x a \ \mathrm d x} x$ Linear Combination of Integrals $\displaystyle$ $=$ $\displaystyle \frac \pi 2 \ln \left\vert{x}\right\vert - \int \frac {\arctan \frac x a \ \mathrm d x} x + C$ Primitive of Reciprocal

$\blacksquare$