Mathematics

$$\int_0^{\pi /2} {\dfrac{1}{{1 + {{\tan }^3}x}}} dx$$


SOLUTION
We have,
$$I=\int _{ 0 }^{ \pi /2 }{ \dfrac { 1 }{ { 1+{ { \tan   }^{ 3 } }x } }  } dx........\left( 1 \right)  \\ $$

Apply property, we get 
$$I=\int _{ 0 }^{ \pi /2 }{ \dfrac { 1 }{ { 1+{ { \cot   }^{ 3 } }x } }  } dx \\ I=\int _{ 0 }^{ \pi /2 }{ \dfrac { { { { \tan   }^{ 3 } }x } }{ { { { \tan   }^{ 3 } }x+1 } }  } dx.........\left( 2 \right) $$

On adding equation (1) and (2), we get
$$ \\ 2I=\int _{ 0 }^{ \pi /2 }{ dx }  \\ I=\dfrac { \pi  }{ 2 }  \\ I=\dfrac { \pi  }{ 4 } $$

Hence, this is the answer.
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Subjective Medium Published on 17th 09, 2020
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