Integration of Secant Squared X

Integration of secant squared of x is an important integral formula in integral calculus; this integral belongs to 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 derivative, so the integral sign \int  {} and \frac{d}{{dx}} on the right side will cancel each other, i.e.

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

Other Integral Formulas of Secant Tangent Function:

The other formulas of secant squared of x integral with angle is in the form of function are given as


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


\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