Secant Integral Formula

In this tutorial we will prove the formula of secant integral which is also an important formula in integral calculus; this integral belongs to the category of trigonometric integral formulae.
The integration of 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 further we can 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}

So 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 Formulas of Secant Function:
The other formulas of secant integral with angle of sine is in the form of function are given as


\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


\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