Definite Integral of Secx Tanx from 0 to Pi over 4

In this tutorial we shall derive definite integral of trigonometric function secant tangent from limits 0 to Pi over 4.

The integration of the form

I =  \int\limits_0^{\frac{\pi }{4}} {\sec x\tan xdx}


First we evaluate this integration by using integral formula \int {\sec x\tan xdx = \sec x} , then we using the basic rule of definite integral \int\limits_a^b {f\left( x \right)dx = \left|  {F\left( x \right)} \right|_a^b}  =  \left[ {F\left( b \right) - F\left( a \right)} \right], we have

\begin{gathered} \int\limits_0^{\frac{\pi }{4}} {\sec x\tan  xdx}  = \left| {\sec x}  \right|_0^{\frac{\pi }{4}} \\ \Rightarrow \int\limits_0^{\frac{\pi }{4}}  {\sec x\tan xdx}  = \left[ {\sec  \frac{\pi }{4} - \sec 0} \right] \\ \end{gathered}


From trigonometric values \sec \frac{\pi }{4} = \sqrt 2 and \sec 0 = 1, we have

\begin{gathered} \Rightarrow \int\limits_0^{\frac{\pi }{4}}  {\sec x\tan xdx}  = \left[ {\sqrt 2  - 1} \right] \\ \Rightarrow \int\limits_0^{\frac{\pi }{4}}  {\sec x\tan xdx}  = \sqrt 2  - 1 \\ \end{gathered}

Comments

comments