# Primitive of x by Arccosecant of x over a

## Theorem

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

## 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:

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

and let:

 $\displaystyle \frac {\d v} {\d x}$ $=$ $\displaystyle x$ $\displaystyle \leadsto \ \$ $\displaystyle v$ $=$ $\displaystyle \frac {x^2} 2$ Primitive of Power

First let $\arccsc \dfrac x a$ be in the interval $\openint 0 {\dfrac \pi 2}$.

Then:

 $\displaystyle \int x \arccsc \frac x a \rd x$ $=$ $\displaystyle \frac {x^2} 2 \arccsc \frac x a - \int \frac {x^2} 2 \paren {\frac {-a} {x \sqrt {x^2 - a^2} } } \rd x + C$ Integration by Parts $\displaystyle$ $=$ $\displaystyle \frac {x^2} 2 \arccsc \frac x a + \frac a 2 \int \frac {x \ \mathrm d x} {\sqrt {x^2 - a^2} } + C$ Primitive of Constant Multiple of Function $\displaystyle$ $=$ $\displaystyle \frac {x^2} 2 \arccsc \frac x a + \frac a 2 \sqrt {x^2 - a^2} + C$ Primitive of $\dfrac x {\sqrt {x^2 - a^2} }$ $\displaystyle$ $=$ $\displaystyle \frac {x^2} 2 \arccsc \frac x a + \frac {a \sqrt{x^2 - a^2} } 2 + C$ simplifying

Similarly, let $\arccsc \dfrac x a$ be in the interval $\openint {-\dfrac \pi 2} 0$.

Then:

 $\displaystyle \int x \arccsc \frac x a \rd x$ $=$ $\displaystyle \frac {x^2} 2 \arccsc \frac x a - \int \frac {x^2} 2 \paren {\frac {-a} {x \sqrt {x^2 - a^2} } } \rd x + C$ Integration by Parts $\displaystyle$ $=$ $\displaystyle \frac {x^2} 2 \arccsc \frac x a - \frac a 2 \int \frac {x \rd x} {\sqrt {x^2 - a^2} } + C$ Primitive of Constant Multiple of Function $\displaystyle$ $=$ $\displaystyle \frac {x^2} 2 \arccsc \frac x a - \frac a 2 \sqrt {x^2 - a^2} + C$ Primitive of $\dfrac x {\sqrt {x^2 - a^2} }$ $\displaystyle$ $=$ $\displaystyle \frac {x^2} 2 \arccsc \frac x a - \frac {a \sqrt{x^2 - a^2} } 2 + C$ simplifying

$\blacksquare$