Mathematics

If $$\displaystyle\int _{ 0 }^{ \pi  }{ xf\left( \sin ^{ 2 }{ x } +\sec ^{ 2 }{ x }  \right) dx } =k \displaystyle\int _{ 0 }^{ { \pi  }/{ 2 } }{ f\left( \sin ^{ 2 }{ x } +\sec ^{ 2 }{ x }  \right) dx } $$, then the value of $$k$$ is


ANSWER

$$\pi $$


SOLUTION
We have, $$\displaystyle\int _{ 0 }^{ \pi  }{ xf\left( \sin ^{ 2 }{ x } +\sec ^{ 2 }{ x }  \right) dx } =k \displaystyle\int _{ 0 }^{ { \pi  }/{ 2 } }{ f\left( \sin ^{ 2 }{ x } +\sec ^{ 2 }{ x }  \right) dx } $$
Let $$I=\displaystyle\int _{ 0 }^{ \pi  }{ xf\left( \sin ^{ 2 }{ x } +\sec ^{ 2 }{ x }  \right) dx } $$               .....(i)
   $$=\displaystyle\int _{ 0 }^{ \pi  }{ \left( \pi -x \right) f\left( \sin ^{ 2 }{ \left( \pi -x \right)  } +\sec ^{ 2 }{ \left( \pi -x \right)  }  \right) dx } $$
   $$=\displaystyle\int _{ 0 }^{ \pi  }{ \left( \pi -x \right) f\left( \sin ^{ 2 }{ x } +\sec ^{ 2 }{ x }  \right) dx } $$                 ....(ii)
On adding equations (i) and (ii), we get
$$2I=\pi \displaystyle\int _{ 0 }^{ \pi  }{ f\left( \sin ^{ 2 }{ x } +\sec ^{ 2 }{ x }  \right) dx } $$
$$\Rightarrow 2I=2\pi \displaystyle\int _{ 0 }^{ { \pi  }/{ 2 } }{ f\left( \sin ^{ 2 }{ x } +\sec ^{ 2 }{ x }  \right) dx } $$
$$\Rightarrow I=\pi \displaystyle\int _{ 0 }^{ { \pi  }/{ 2 } }{ f\left( \sin ^{ 2 }{ x } +\sec ^{ 2 }{ x }  \right) dx } $$
On comparing with given integral, we get $$k=\pi $$
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Single Correct Medium Published on 17th 09, 2020
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