Mathematics

Evaluate: $$\int { \dfrac { { x }^{ 2 }dx }{ ({ x }^{ 2 }+1)({ x }^{ 2 }+2) }  } $$


ANSWER

$$\sqrt { 2 } \tan ^{ -1 }{ \frac { x }{ \sqrt { 2 } } -\tan ^{ -1 }{ x } +c } $$


SOLUTION
Given,

$$\int \dfrac{x^2}{\left(x^2+1\right)\left(x^2+2\right)}dx$$

using partial fraction, we get,

$$=\int \dfrac{2}{x^2+2}-\dfrac{1}{x^2+1}dx$$

$$=\int \dfrac{2}{x^2+2}dx-\int \dfrac{1}{x^2+1}dx$$

$$=\sqrt{2}\tan ^{-1}\left(\dfrac{x}{\sqrt{2}}\right)-\tan ^{-1}\left(x\right)+C$$
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Single Correct Medium Published on 17th 09, 2020
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