# Cotangent Integral Formula

In this tutorial we will prove the formula of cotangent integral which is also an important formula in integral calculus; this integral belongs to trigonometric formulae.
The integration of cotangent function is of the form

To prove this formula, consider

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

Alternate Proof:
We have given integration of the form

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 Formulas of Cotangent Function:
The other formulas of cotangent integral with angle of sine is in the form of 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