# Tangent Integral Formula

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 $\frac{d}{{dx}}\ln x = \frac{1}{x}$ 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 derivative, so the integral sign $\int {}$and $\frac{d}{{dx}}$ 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 $f\left( x \right) = \cos x$ 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 Formulas of Tangent Function:

The other formulas of tangent integral with angle of sine is in the form of 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

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