Mathematics

The value of $$\int\dfrac{{x^{2}}}{{(x^{2}+a^{2})(x^{2}+b^{2})}}dx$$ is


ANSWER

$$none\ of\ these$$


SOLUTION
$$\displaystyle \int { \dfrac { { x }^{ 2 } }{ \left( { x }^{ 2 }+a^{ 2 } \right) \left( { x }^{ 2 }+b^{ 2 } \right)  } dx } =\displaystyle \int { \dfrac { { x }^{ 2 }+{ a }^{ 2 }-{ a }^{ 2 } }{ \left( { x }^{ 2 }+a^{ 2 } \right) \left( { x }^{ 2 }+b^{ 2 } \right)  } dx } $$
$$I=\displaystyle \int { \dfrac { dx }{ \left( { x }^{ 2 }+b^{ 2 } \right)  }  } -\displaystyle \int { \dfrac { { a }^{ 2 } }{ \left( { x }^{ 2 }+a^{ 2 } \right) \left( { x }^{ 2 }+b^{ 2 } \right)  } dx } $$
$${ I }_{ 1 }={ a }^{ 2 }\displaystyle \int { \dfrac { dx }{ \left( { x }^{ 2 }+a^{ 2 } \right) \left( { x }^{ 2 }+b^{ 2 } \right)  }  } $$
$${ I }_{ 1 }=\displaystyle \int { \dfrac { { a }^{ 2 } }{ \left( { x }^{ 2 }+a^{ 2 } \right) -\left( { x }^{ 2 }+b^{ 2 } \right)  }  } \displaystyle \int { \dfrac { \left( { x }^{ 2 }+a^{ 2 } \right) -\left( { x }^{ 2 }+b^{ 2 } \right)  }{ \left( { x }^{ 2 }+a^{ 2 } \right) \left( { x }^{ 2 }+b^{ 2 } \right)  }  } dx$$
$${ I }_{ 1 }=\dfrac { { a }^{ 2 } }{ { a }^{ 2 }-{ b }^{ 2 } } \displaystyle \int { \dfrac { \left( { x }^{ 2 }+a^{ 2 } \right) dx }{ \left( { x }^{ 2 }+a^{ 2 } \right) \left( { x }^{ 2 }+b^{ 2 } \right)  }  } -\dfrac { { a }^{ 2 } }{ { a }^{ 2 }-{ b }^{ 2 } } \displaystyle \int { \dfrac { { x }^{ 2 }+{ b }^{ 2 }dx }{ \left( { a }^{ 2 }+x^{ 2 } \right) \left( { x }^{ 2 }+b^{ 2 } \right)  }  } $$
$${ I }_{ 1 }=\dfrac { { a }^{ 2 } }{ { a }^{ 2 }-{ b }^{ 2 } } \displaystyle \int { \dfrac { dx }{ { x }^{ 2 }-{ b }^{ 2 } }  } \dfrac { { a }^{ 2 } }{ { a }^{ 2 }-{ b }^{ 2 } } \displaystyle \int { \dfrac { dx }{ { x }^{ 2 }-{ a }^{ 2 } }  } $$
$${ I }_{ 1 }=\dfrac { { a }^{ 2 } }{ { a }^{ 2 }-{ b }^{ 2 } } \left\{ \dfrac { 1 }{ 2 } \tan ^{ -1 }{ \left( \dfrac { x }{ b }  \right)  } -\dfrac { 1 }{ a } \tan ^{ -1 }{ \left( \dfrac { x }{ a }  \right)  }  \right\} $$
$$I=\dfrac { 1 }{ b } \tan ^{ -1 }{ \left( \dfrac { x }{ a }  \right)  } -\dfrac { { a }^{ 2 } }{ { a }^{ 2 }-{ b }^{ 2 } } \left[ \dfrac { 1 }{ b } \tan ^{ -1 }{ \left( \dfrac { x }{ b }  \right)  } -\dfrac { 1 }{ b } \tan ^{ -1 }{ \dfrac { x }{ a }  }  \right] $$

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Single Correct Medium Published on 17th 09, 2020
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