# Primitive of Function of Arccotangent

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## Theorem

$\ds \int \map F {\arccot \frac x a} \rd x = -a \int \map F u \csc^2 u \rd u$

where $u = \arccot \dfrac x a$.

## Proof

First note that:

 $\ds u$ $=$ $\ds \arccot \frac x a$ $\ds \leadsto \ \$ $\ds x$ $=$ $\ds a \cot u$ Definition of Arccotangent

Then:

 $\ds u$ $=$ $\ds \arccot \frac x a$ $\ds \leadsto \ \$ $\ds \frac {\d u} {\d x}$ $=$ $\ds \frac {-a} {a^2 + x^2}$ Derivative of Arccotangent Function: Corollary $\ds \leadsto \ \$ $\ds \int \map F {\arccot \frac x a} \rd x$ $=$ $\ds \int \map F u \frac {a^2 + x^2} {-a} \rd u$ Primitive of Composite Function $\ds$ $=$ $\ds \int \map F u \frac {a^2 + a^2 \cot^2 u} {-a} \rd u$ Definition of $x$ $\ds$ $=$ $\ds \int \map F u a^2 \frac {1 + \cot^2 u} {-a} \rd u$ $\ds$ $=$ $\ds \int \map F u \paren {-a} \csc^2 u \rd u$ Difference of Squares of Cosecant and Cotangent $\ds$ $=$ $\ds -a \int \map F u \csc^2 u \rd u$ Primitive of Constant Multiple of Function

$\blacksquare$