Secant Integral Formula

In this tutorial we will prove the formula of secant integral which is another important formula in integral calculus. This integral belongs to the category of trigonometric integral formulae.

The integration of the secant function is of the form
\[\int {\sec xdx = } \ln \left( {\sec x + \tan x} \right) + c = \ln \tan \left( {\frac{x}{2} + \frac{\pi }{2}} \right) + c\]

To prove this formula, consider
\[\int {\sec x} dx = \int {\frac{{\sec x\left( {\sec x + \tan x} \right)}}{{\sec x + \tan x}}} dx\]

By multiplying and dividing the relation $$\left( {\sec x + \tan x} \right)$$
\[\begin{gathered} \int {\sec x} dx = \int {\frac{{{{\sec }^2}x + \sec x\tan x}}{{\sec x + \tan x}}} dx \\ \Rightarrow \int {\sec x} dx = \int {\frac{{\sec x\tan x + {{\sec }^2}x}}{{\sec x + \tan x}}} dx \\ \end{gathered} \]

Here $$f\left( x \right) = \sec x + \tan x$$, then $$f’\left( x \right) = \sec x\tan x + {\sec ^2}x$$

Now using the formula of integration $$\int {\frac{{f’\left( x \right)}}{{f\left( x \right)}}dx = \ln f\left( x \right) + c} $$, we have
\[\int {\sec x} dx = \ln \left( {\sec x + \tan x} \right) + c\]

Now we can further solve this result as follows:
\[\begin{gathered} \int {\sec x} dx = \ln \left( {\frac{1}{{\cos x}} + \frac{{\sin x}}{{\cos x}}} \right) + c = \ln \left( {\frac{{1 + \sin x}}{{\cos x}}} \right) + c \\ \Rightarrow \int {\sec x} dx = \ln \left[ {\frac{{1 – \cos \left( {x + \frac{\pi }{2}} \right)}}{{\sin \left( {x + \frac{\pi }{2}} \right)}}} \right] + c \\ \Rightarrow \int {\sec x} dx = \ln \left[ {\frac{{2{{\sin }^2}\left( {\frac{x}{2} + \frac{\pi }{2}} \right)}}{{2\sin \left( {\frac{x}{2} + \frac{\pi }{2}} \right)\cos \left( {\frac{x}{2} + \frac{\pi }{2}} \right)}}} \right] + c \\ \Rightarrow \int {\sec x} dx = \ln \left[ {\frac{{\sin \left( {\frac{x}{2} + \frac{\pi }{2}} \right)}}{{\cos \left( {\frac{x}{2} + \frac{\pi }{2}} \right)}}} \right] + c \\ \Rightarrow \int {\sec x} dx = \ln \tan \left( {\frac{x}{2} + \frac{\pi }{2}} \right) + c \\ \end{gathered} \]

In conclusion we can write this formula as
\[\int {\sec xdx = } \ln \left( {\sec x + \tan x} \right) + c = \ln \tan \left( {\frac{x}{2} + \frac{\pi }{2}} \right) + c\]

Other Integral Formulae of the Secant Function

The other formulae of secant integral with an angle of sine in the form of a function are given as

1. \[\int {\sec axdx = } \frac{1}{a}\ln \left( {\sec ax + \tan ax} \right) + c = \frac{1}{a}\ln \tan \left( {\frac{{ax}}{2} + \frac{\pi }{2}} \right) + c\]

2. \[\int {\sec f\left( x \right)f’\left( x \right)dx = \ln \left[ {\sec f\left( x \right) + \tan f\left( x \right)} \right] + c} = \ln \tan \left[ {\frac{{f\left( x \right)}}{2} + \frac{\pi }{2}} \right] + c\]