Mathematics

Evaluate
$$\displaystyle \int _{ 0 }^{ \pi /2 }{ \dfrac { x+\sin { x }  }{ 1+\cos { x }  } dx } $$


SOLUTION
$$\begin{array}{l} \int  _{ 0 }^{ \frac { \pi  }{ 2 }  }{ \frac { { x+\sin  x } }{ { 1+\cos  x } } dx } \\ =\, \, \int  _{ 0 }^{ \frac { \pi  }{ 2 }  }{ \frac { { x+2\sin  \frac { x }{ 2 } \cos  \frac { x }{ 2 }  } }{ { 2{ { \cos   }^{ 2 } }\frac { x }{ 2 }  } } dx } \\ =\, \int  _{ 0 }^{ \frac { \pi  }{ 2 }  }{ \left( { x\frac { 1 }{ 2 } { { \sec   }^{ 2 } }\frac { x }{ 2 } +\tan  \frac { x }{ 2 }  } \right)  }\, \, dx \\ \therefore \, \, \, from\, formula\, ,\int { \left[ { x\, { f^{ ' } }(x)+f(x) } \right]  } \, \, dx=x\, f(x) \\ =\, \left[ { x\tan  \frac { x }{ 2 }  } \right] _{ 0 }^{ \frac { \pi  }{ 2 }  } \\ =\, \left( { \frac { \pi  }{ 2 } \tan  \frac { \pi  }{ 4 }  } \right) -0 \\ =\frac { \pi  }{ 2 }  \end{array}$$
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Subjective Medium Published on 17th 09, 2020
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