# Integration of Cscx Cotx

Integration of cosecant cotangent function is an important integral formula in integral calculus; this integral belongs to trigonometric formulae.

The integration of cosecant cotangent is of the form

To prove this formula, consider

Using the derivative formula $\frac{d}{{dx}}\csc x = - \csc x\cot x$, we have

Integrating both sides of equation (i) with respect to $x$, we have

As we know that by definition integration is the inverse process of derivative, so the integral sign $\int {}$and $\frac{d}{{dx}}$ on the right side will cancel each other, i.e.

Other Integral Formulas of Cosecant Cotangent Function:
The other formulas of cosecant tangent integral with angle is in the form of function are given as

1.

2.

Example: Evaluate the integral $\int {\csc 9x\cot 9xdx}$ with respect to $x$

We have integral

Using the formula $\int {\csc ax\cot axdx = - \frac{{\csc ax}}{a}} + c$, we have