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.