In this tutorial we will prove the formula of tangent integral which is also an important formula in integral calculus; this integral belongs to trigonometric formulae.

The integration of tangent function is of the form

or

To prove this formula, consider

Using the derivative formulas and , we have

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

As we know that by definition integration is the inverse process of derivative, so the integral sign and on the right side will cancel each other, i.e.

Further we can prove this formula in another form as,

__Alternate Proof__**:**

We have given integration of the form

Here we have then

Using the formula of integration,

__Other Integral Formulas of Tangent Function__**:**

The other formulas of tangent integral with angle of sine is in the form of function are given as

**1. **** **

**2. **** **

**3. **** **

**4. **** **

__Example__**:** Evaluate the integral with respect to

We have integral

Using the formula , we have