Mathematics

$$\displaystyle\int_{0}^{\pi}\frac{dx}{1+a^{\cos x}} $$ equals


ANSWER

$$\displaystyle\frac{\pi }{2}$$


SOLUTION
$$\displaystyle I=\int _{ 0 }^{ \pi  } \frac { dx }{ 1+a^{ \cos  x } } =\int _{ 0 }^{ \pi  } \frac { dx }{ 1+a^{ \cos  \left( \pi -x \right)  } } =\int _{ 0 }^{ \pi  } \frac { dx }{ 1+a^{ -\cos  x } } =\int _{ 0 }^{ \pi  } \left( 1-\frac { 1 }{ 1+a^{ \cos  x } }  \right) dx$$
$$\Rightarrow I=\pi -I$$
$$\Rightarrow I=\dfrac { \pi  }{ 2 } $$

Ans: B
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Single Correct Medium Published on 17th 09, 2020
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