Tangent Integral Formula

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

The integration of tangent function is of the form

or

To prove this formula, consider

Using the derivative formulas $\frac{d}{{dx}}\ln x = \frac{1}{x}$ and $\frac{d}{{dx}}\sec x = \sec \tan x$, 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.

We can further prove this formula in another form as

Alternate Proof

We have the integration of the form

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

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 Tangent Function

The other formulae of tangent integral with an angle of sine in the form of a function are given as

1. $\int {\tan axdx = \frac{1}{a}\ln \sec ax + c}$

2. $\int {\tan axdx = - \frac{1}{a}\ln \cos ax + c}$

3. $\int {\tan f\left( x \right)f'\left( x \right)dx = \ln \sec f\left( x \right) + c}$

4. $\int {\tan f\left( x \right)f'\left( x \right)dx = - \ln \cos f\left( x \right) + c}$

Example: Evaluate the integral $\int {\tan 6xdx}$ with respect to $x$

We have integral

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