# Primitive of Arccosecant of x over a

## Theorem

$\displaystyle \int \operatorname{arccsc} \frac x a \rd x = \begin{cases} \displaystyle x \operatorname{arccsc} \frac x a + a \ln \left({x + \sqrt {x^2 - a^2} }\right) + C & : 0 < \operatorname{arccsc} \dfrac x a < \dfrac \pi 2 \\ \displaystyle x \operatorname{arccsc} \frac x a - a \ln \left({x + \sqrt {x^2 - a^2} }\right) + C & : -\dfrac \pi 2 < \operatorname{arccsc} \dfrac x a < 0 \\ \end{cases}$

## Proof

With a view to expressing the primitive in the form:

$\displaystyle \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$

let:

 $\displaystyle u$ $=$ $\displaystyle \operatorname{arccsc} \frac x a$ $\displaystyle \implies \ \$ $\displaystyle \frac {\d u} {\d x}$ $=$ $\displaystyle \begin{cases} \dfrac {-a} {x \sqrt {x^2 - a^2} } & : 0 < \operatorname{arccsc} \dfrac x a < \dfrac \pi 2 \\ \dfrac a {x \sqrt {x^2 - a^2} } & : -\dfrac \pi 2 < \operatorname{arccsc} \dfrac x a < 0 \\ \end{cases}$ Derivative of $\operatorname{arccsc} \dfrac x a$

and let:

 $\displaystyle \frac {\d v} {\d x}$ $=$ $\displaystyle 1$ $\displaystyle \implies \ \$ $\displaystyle v$ $=$ $\displaystyle x$ Primitive of Constant

First let $\operatorname{arccsc} \dfrac x a$ be in the interval $\left({0 \,.\,.\,\dfrac \pi 2}\right)$.

Then:

 $\displaystyle \int \operatorname{arccsc} \frac x a \rd x$ $=$ $\displaystyle x \operatorname{arccsc} \frac x a - \int x \left({\frac {-a} {x \sqrt {x^2 - a^2} } }\right) \rd x + C$ Integration by Parts $\displaystyle$ $=$ $\displaystyle x \operatorname{arccsc} \frac x a + a \int \frac {\d x} {\sqrt {x^2 - a^2} } + C$ Primitive of Constant Multiple of Function $\displaystyle$ $=$ $\displaystyle x \operatorname{arccsc} \frac x a + a \ln \left({x + \sqrt {x^2 - a^2} }\right) + C$ Primitive of $\dfrac 1 {\sqrt {x^2 - a^2} }$

Similarly, let $\operatorname{arccsc} \dfrac x a$ be in the interval $\left({-\dfrac \pi 2 \,.\,.\, 0}\right)$.

Then:

 $\displaystyle \int \operatorname{arccsc} \frac x a \rd x$ $=$ $\displaystyle x \operatorname{arccsc} \frac x a - \int x \left({\frac a {x \sqrt {x^2 - a^2} } }\right) \rd x + C$ Integration by Parts $\displaystyle$ $=$ $\displaystyle x \operatorname{arccsc} \frac x a - a \int \frac {\d x} {\sqrt {x^2 - a^2} } + C$ Primitive of Constant Multiple of Function $\displaystyle$ $=$ $\displaystyle x \operatorname{arccsc} \frac x a - a \ln \left({x + \sqrt {x^2 - a^2} }\right) + C$ Primitive of $\dfrac 1 {\sqrt {x^2 - a^2} }$

$\blacksquare$