# Primitive of Function of Arccosecant

## Theorem

$\displaystyle \int F \left({\operatorname{arccsc} \frac x a}\right) \ \mathrm d x = -a \int F \left({u}\right) \left\vert{\csc u}\right\vert \cot u \ \mathrm d u$

where $u = \operatorname{arccsc} \dfrac x a$.

## Proof

First note that:

 $\displaystyle u$ $=$ $\displaystyle \operatorname{arccsc} \frac x a$ $\displaystyle \implies \ \$ $\displaystyle x$ $=$ $\displaystyle a \csc u$ Definition of Arccosecant

Then:

 $\displaystyle u$ $=$ $\displaystyle \operatorname{arccsc} \frac x a$ $\displaystyle \implies \ \$ $\displaystyle \frac{\mathrm d u} {\mathrm d x}$ $=$ $\displaystyle \frac {-a} {\left\vert{x}\right\vert {\sqrt {x^2 - a^2} } }$ Derivative of Arccosecant Function: Corollary $\displaystyle \implies \ \$ $\displaystyle \int F \left({\operatorname{arccsc} \frac x a}\right) \ \mathrm d x$ $=$ $\displaystyle \int F \left({u}\right) \ \frac {\left\vert{x}\right\vert {\sqrt {x^2 - a^2} } } {-a} \ \mathrm d u$ Primitive of Composite Function $\displaystyle$ $=$ $\displaystyle \int F \left({u}\right) \ \frac {\left\vert{a \csc u}\right\vert {\sqrt {a^2 \csc^2 u - a^2} } } {-a} \ \mathrm d u$ Definition of $x$ $\displaystyle$ $=$ $\displaystyle \int F \left({u}\right) \ \left\vert{\csc u}\right\vert \left({-\sqrt {a^2 \csc^2 u - a^2} }\right) \ \mathrm d u$ $\displaystyle$ $=$ $\displaystyle \int F \left({u}\right) \ \left({-a}\right) \left\vert{\csc u}\right\vert {\sqrt {\csc^2 u - 1} } \ \mathrm d u$ $\displaystyle$ $=$ $\displaystyle \int F \left({u}\right) \ \left({-a}\right) \left\vert{\csc u}\right\vert \cot u \ \mathrm d u$ Difference of Squares of Cosecant and Cotangent $\displaystyle$ $=$ $\displaystyle -a \int F \left({u}\right) \ \left\vert{\csc u}\right\vert \cot u \ \mathrm d u$ Primitive of Constant Multiple of Function

$\blacksquare$