Definite Integral of Tanx from 0 to Pi over 4

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

The integration of the form

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


First we evaluate this integration by using integral formula \int {\tan xdx =  - \ln \cos 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}} {\tan  xdx}  = \left| { - \ln \cos x}  \right|_0^{\frac{\pi }{4}} \\ \Rightarrow \int\limits_0^{\frac{\pi }{4}}  {\tan xdx}  =  - \left| {\ln \cos x} \right|_0^{\frac{\pi  }{4}} \\ \Rightarrow \int\limits_0^{\frac{\pi }{4}}  {\tan xdx}  =  - \left[ {\ln \cos \frac{\pi }{4} - \ln \cos  0} \right] \\ \end{gathered}


From trigonometric values

\cos 0 = 1

and \cos \frac{\pi }{4} = \frac{1}{{\sqrt 2 }}, we have

\Rightarrow  \int\limits_0^{\frac{\pi }{4}} {\tan xdx}   =  - \left[ {\ln \frac{1}{{\sqrt 2  }} - \ln 1} \right]


Using the value \ln 1 =  0, we have

\begin{gathered} \Rightarrow \int\limits_0^{\frac{\pi }{4}}  {\tan xdx}  =  - \left[ {\ln \frac{1}{{\sqrt 2 }} - 0}  \right] \\ \Rightarrow \int\limits_0^{\frac{\pi }{4}}  {\tan xdx}  = \ln {\left(  {\frac{1}{{\sqrt 2 }}} \right)^{ - 1}} = \ln \sqrt 2 \\ \Rightarrow \int\limits_0^{\frac{\pi }{4}}  {\tan xdx}  = \ln {2^{\frac{1}{2}}} =  \frac{1}{2}\ln 2 \\ \end{gathered}

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