Mathematics

Solve:
$$\displaystyle\int\dfrac{1}{{\left( {{x^2} + 1} \right)\left( {{x^2} + 4} \right)}}dx$$


SOLUTION
$$I=\displaystyle \int \dfrac {dx}{(x^2+1)(x^2+4)}=\displaystyle \int \left (\dfrac {1}{3}\right)\times 3\ dx$$
$$=\dfrac {1}{3}\dfrac {[(x^2+4)-(x^2+1)]dx}{(x^2+1)(x^2+4)}$$
$$\Rightarrow \ \dfrac {1}{3}\displaystyle \int \dfrac {dx}{(xz^-4)}-\dfrac {1}{3} \ \displaystyle \int \dfrac {dx}{x^2+2^2}$$
$$\Rightarrow \ \dfrac {1}{3} \left [\tan^2x-\dfrac {1}{2}- \tan ^{-1}\dfrac {x}{2}\right]+c$$
$$I\Rightarrow \ \dfrac {1}{6} [2\tan^2x-\tan^{-1}x/2]+c$$

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Subjective Medium Published on 17th 09, 2020
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