# Primitive of Function of Arccosecant

## Theorem

$\ds \int \map F {\arccsc \frac x a} \rd x = -a \int \map F u \size {\csc u} \cot u \rd u$

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

## Proof

First note that:

 $\ds u$ $=$ $\ds \arccsc \frac x a$ $\ds \leadsto \ \$ $\ds x$ $=$ $\ds a \csc u$ Definition of Arccosecant

Then:

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

$\blacksquare$