Mathematics

Evaluate the integrals using substitution.
$$\displaystyle\int^1_0\dfrac{x^2}{x^2+1}dx$$.


SOLUTION
$$\int _{ 0 }^{ 12 }{ \cfrac { { x }^{ 2 } }{ { x }^{ 2 }+1 }  } dx$$
$$x=\tan { \theta  } \Rightarrow dx=\sec ^{ 2 }{ \theta  } d\theta \quad $$
$$x=0\Rightarrow \theta ={ 0 }^{ o },x=1\Rightarrow \theta =\cfrac { \pi  }{ 4 } $$
$$\quad =\int _{ 0 }^{ \pi /4 }{ \cfrac { \tan ^{ 2 }{ \theta  }  }{ \sec ^{ 2 }{ \theta  }  } .\sec ^{ 2 }{ \theta  } .d\theta  } =\int _{ 0 }^{ \pi /4 }{ \tan ^{ 2 }{ \theta  }  } .d\theta $$
$$=\int _{ 0 }^{ \pi /4 }{ \left( \sec ^{ 2 }{ \theta  } -1 \right)  } .d\theta ={ \left[ \tan { \theta  } -\theta  \right]  }_{ 0 }^{ \pi /4 }=1-\cfrac { \pi  }{ 4 } $$
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Subjective Medium Published on 17th 09, 2020
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