Integral of Inverse Cosecant

Integral of inverse cosecant ${\csc ^{ - 1}}x$ it is an important integral function, but it has no direct method to find we shall find the integration of cosecant inverse by using integration by parts method.

The integral of cosecant inverse is of the form

To solve this integration, it must have at least two functions, it in this function it has only one function that is ${\csc ^{ - 1}}x$, now consider second function as $1$. Now the integration becomes

Take first function is ${\csc ^{ - 1}}x$ and second function will be $1$
Using formula for integration by parts, we have

Equation (i) becomes using above formula, we have

It can be written in the form, we get

Using formula

we have

Now further we can use this integration of cosecant inverse as a formula.