Integration of Secx Tanx

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

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

Other Integral Formulas of Secant Tangent Function:
The other formulas of secant tangent integral with angle is in the form of function are given as


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


\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