Mathematics

The value of $$\int _{ 0 }^{ \pi /4 }{ \dfrac { 1+\tan { x }  }{ 1-\tan { x }  }  } dx$$ is


SOLUTION
$$\begin{array}{l} \int _{ 0 }^{ \frac { \pi  }{ 4 }  }{ \frac { { \left( { 1+\tan  x } \right)  } }{ { \left( { 1-\tan  x } \right)  } } dx }  \\ \Rightarrow \int _{ 0 }^{ \frac { \pi  }{ 4 }  }{ \tan  \left( { \frac { \pi  }{ 4 } +x } \right) dx }  \\ \Rightarrow -\left[ { \log  \left| { \cos  \left( { \frac { \pi  }{ 4 } +x } \right)  } \right|  } \right] _{ 0 }^{ \frac { \pi  }{ 4 }  }+c \\ \Rightarrow -\log  \left| { \cos  \frac { \pi  }{ 2 } -\cos  \frac { \pi  }{ 4 }  } \right| +c \\ \Rightarrow -\log  \left| { -\frac { 1 }{ { \sqrt { 2 }  } }  } \right| +c \\ \Rightarrow -\log { 2^{ \frac { { -1 } }{ 2 }  } } +c \\ \frac { 1 }{ 2 } { \log { 2 }  }+c \end{array}$$
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Subjective Medium Published on 17th 09, 2020
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