Mathematics

$$\displaystyle \int_{\pi /4}^{3\pi /4}\frac{dx}{1+\cos x}$$ is equal to


ANSWER

$$2$$


SOLUTION
Let $$\displaystyle I=\int { \dfrac { 1 }{ 1+\cos { x }  }  } dx$$

Substitute $$\displaystyle t=\tan { \dfrac { x }{ 2 }  } \Rightarrow dt=\dfrac { 1 }{ 2 } dxsec^{ 2 }x$$

$$\displaystyle I=\int { \dfrac { 2 }{ \left( { u }^{ 2 }+1 \right) \left( \dfrac { 1-{ u }^{ 2 } }{ { u }^{ 2 }+1 } +1 \right)  }  } du=\int { du } =u=\tan { \dfrac { x }{ 2 }  } $$

$$\displaystyle \therefore \int _{ \dfrac { \pi  }{ 4 }  }^{ \dfrac { 3\pi  }{ 4 }  }{ \dfrac { dx }{ 1+\cos { x }  }  } =\left[ \tan { \dfrac { x }{ 2 }  }  \right] _{ \dfrac { \pi  }{ 4 }  }^{ \dfrac { 3\pi  }{ 4 }  }=2$$
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Single Correct Medium Published on 17th 09, 2020
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