Primitive of Arccosecant of x over a

Theorem

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

x \arccsc \dfrac x a + a \map \ln {x + \sqrt {x^2 - a^2} } + C & : 0 < \arccsc \dfrac x a < \dfrac \pi 2 \\ x \arccsc \dfrac x a - a \map \ln {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:
 * $\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$

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 $\arcsin \dfrac x a$


 * Primitive of $\arccos \dfrac x a$


 * Primitive of $\arctan \dfrac x a$


 * Primitive of $\arccot \dfrac x a$


 * Primitive of $\arcsec \dfrac x a$