Primitive of Power of x by Arccosecant of x over a

Theorem

 * $\ds \int x^m \arccsc \frac x a \rd x = \begin{cases}

\dfrac {x^{m + 1} } {m + 1} \arccsc \dfrac x a + \frac a {m + 1} \int \dfrac {x^m \rd x} {\sqrt {x^2 - a^2} } + C & : 0 < \arccsc \dfrac x a < \dfrac \pi 2 \\ \dfrac {x^{m + 1} } {m + 1} \arccsc \dfrac x a - \frac a {m + 1} \int \dfrac {x^m \rd x} {\sqrt {x^2 - a^2} } + C & : -\dfrac \pi 2 < \arccsc \dfrac x a < 0 \\ \end{cases}$

Proof
With a view to expressing the primitive in the form:
 * $\ds \int u \frac {\rd v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$

let:

and let:

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

Then:

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

Then:

Also see

 * Primitive of $x^m \arcsin \dfrac x a$


 * Primitive of $x^m \arccos \dfrac x a$


 * Primitive of $x^m \arctan \dfrac x a$


 * Primitive of $x^m \arccot \dfrac x a$


 * Primitive of $x^m \arcsec \dfrac x a$