Integration of Secant Squared X

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

The integration of secant squared of x is of the form

$\int {{{\sec }^2}xdx = } \tan x + c$

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

Using the derivative formula $\frac{d}{{dx}}\tan x = {\sec ^2}x$, we have
$\begin{gathered} \frac{d}{{dx}}\left[ {\tan x + c} \right] = {\sec ^2}x + 0 \\ \Rightarrow \frac{d}{{dx}}\left[ {\tan x + c} \right] = {\sec ^2}x \\ \Rightarrow {\sec ^2}x = \frac{d}{{dx}}\left[ {\tan x + c} \right] \\ \Rightarrow {\sec ^2}xdx = d\left[ {\tan x + c} \right]\,\,\,\,{\text{ – – – }}\left( {\text{i}} \right) \\ \end{gathered}$

Integrating both sides of equation (i) with respect to $x$, we have
$\int {{{\sec }^2}xdx} = \int {d\left[ {\tan x + c} \right]}$

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.
$\int {{{\sec }^2}xdx = } \tan x + c$

Other Integral Formulae of the Secant Tangent Function

The other formulae of secant squared of x integral with an angle in the form of s function are given as

1. $\int {{{\sec }^2}axdx = \frac{{\tan ax}}{a}} + c$

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

Example: Evaluate the integral $\int {{{\sec }^2}3xdx}$ with respect to $x$

We have integral $I = \int {{{\sec }^2}3xdx}$

Using the formula $\int {{{\sec }^2}axdx = \frac{{\tan ax}}{a}} + c$, we have
$I = \int {{{\sec }^2}3x} = \frac{{\tan 3x}}{3} + c$