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
\[I = \int {{{\sec }^2}3x} = \frac{{\tan 3x}}{3} + c\]