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
$\int {\tan xdx = } \ln \sec x + c$ or $\int {\tan xdx = – } \ln \cos x + c$

To prove this formula, consider
$\frac{d}{{dx}}\left[ {\ln \sec x + c} \right] = \frac{d}{{dx}}\ln \sec x + \frac{d}{{dx}}c$

Using the derivative formulas $\frac{d}{{dx}}\ln x = \frac{1}{x}$ and $\frac{d}{{dx}}\sec x = \sec \tan x$, we have
$\begin{gathered} \frac{d}{{dx}}\left[ {\ln \sec x + c} \right] = \frac{1}{{\sec x}}\frac{d}{{dx}}\sec x + 0 \\ \Rightarrow \frac{d}{{dx}}\left[ {\ln \sec x + c} \right] = \frac{1}{{\sec x}}\left( {\sec x\tan x} \right) \\ \Rightarrow \frac{d}{{dx}}\left[ {\ln \sec x + c} \right] = \tan x \\ \Rightarrow \tan x = \frac{d}{{dx}}\left[ {\ln \sec x + c} \right] \\ \tan xdx = d\left[ {\ln \sec x + c} \right]\,\,\,\,{\text{ – – – }}\left( {\text{i}} \right) \\ \end{gathered}$

Integrating both sides of equation (i) with respect to $x$, we have
$\int {\tan xdx} = \int {d\left[ {\ln \sec x + c} \right]}$

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.
$\int {\tan xdx = } \ln \sec x + c$

We can further prove this formula in another form as
$\begin{gathered} \int {\tan xdx = } \ln \sec x + c \\ \Rightarrow \int {\tan xdx = } \ln \frac{1}{{\cos x}} + c \\ \Rightarrow \int {\tan xdx = } \ln {\left( {\cos x} \right)^{ – 1}} + c \\ \Rightarrow \int {\tan xdx = } – \ln \left( {\cos x} \right) + c \\ \end{gathered}$

Alternate Proof

We have the integration of the form
$\int {\tan xdx = \int {\frac{{\sin x}}{{\cos x}}dx} }$

Here we have $f\left( x \right) = \cos x$ then $f’\left( x \right) = – \sin x$
$\int {\tan xdx = – \int {\frac{{ – \sin x}}{{\cos x}}dx} }$

Using the formula of integration, $\int {\frac{{f’\left( x \right)}}{{f\left( x \right)}}dx = \ln f\left( x \right) + c}$
$\int {\tan xdx = – \ln \cos x + c}$

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 $I = \int {\tan 6xdx}$

Using the formula $\int {\tan axdx = \frac{1}{a}\ln \sec ax + c}$, we have
$\int {\tan 6xdx} = \frac{1}{6}\ln \sec 6x + c$