# Cotangent Integral Formula

In this tutorial we will prove the formula of cotangent integral which is another important formula in integral calculus. This integral belongs to the trigonometric formulae.

The integration of the cotangent function is of the form

To prove this formula, consider

Using the derivative formula $\frac{d}{{dx}}\ln x = \frac{1}{x}$ and $\frac{d}{{dx}}\sin x = \cos x$, 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 the derivative, the integral sign $\int {}$and $\frac{d}{{dx}}$ on the right side will cancel each other out, i.e.

Alternate Proof

We substitute $\cot x$  with $\frac{{\cos x}}{{\sin x}}$

Here we have $f\left( x \right) = \sin x$ then $f'\left( x \right) = - \cos x$

Using the formula of integration, $\int {\frac{{f'\left( x \right)}}{{f\left( x \right)}}dx = \ln f\left( x \right) + c}$

Other Integral Formulae of the Cotangent Function
The other formulae of cotangent integral with an angle of sine in the form of a function are given as

1. $\int {\cot axdx = \frac{1}{a}\ln \sin ax + c}$

2. $\int {\cot f\left( x \right)f'\left( x \right)dx = \ln \sin f\left( x \right) + c}$

Example: Evaluate the integral $\int {\cot 3xdx}$ with respect to $x$

We have integral

Using the formula $\int {\cot axdx = \frac{1}{a}\ln \sin ax + c}$, we have