Primitive of Arccotangent of x over a over x

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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

$\blacksquare$


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