Integration of Secx Tanx

Integration of the secant tangent function is an important integral formula in integral calculus, and this integral belongs to the trigonometric formulae.

The integration of secant tangent is of the form
\[\int {\sec x\tan xdx = } \sec x + c\]

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

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

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

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.
\[\int {\sec x\tan xdx = } \sec x + c\]

Other Integral Formulae of the Secant Tangent Function

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

1. \[\int {\sec ax\tan axdx = \frac{{\sec ax}}{a}} + c\]

2. \[\int {\sec f\left( x \right)\tan f\left( x \right)f’\left( x \right)dx = \sec f\left( x \right) + c} \]

Example: Evaluate the integral $$\int {\sec 5x\tan 5xdx} $$ with respect to $$x$$

We have integral \[I = \int {\sec 5x\tan 5xdx} \]

Using the formula $$\int {\sec ax\tan axdx = \frac{{\sec ax}}{a}} + c$$, we have
\[I = \int {\sec 5x\tan 5xdx} = \frac{{\sec 5x}}{5} + c\]