Evaluate: 2x.tan-1(x^ 2 ) 1 + x^ 4 dx. - Vidyadip

Question

Evaluate:

$\int\frac{\text{2x.tan-1(x}^{2})}{\text{1 + x}^{4}}\text{dx}. $

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Answer

$\int\frac{\text{2x.tan}^{-1}\text{(x}^{2})}{\text{1 + x}^{4}}\text{dx}=\int\text{t dt }\text{ where tan}^{-1}\text{(x}^{2})=\text{t }\text{ }\therefore\frac{\text{2x}}{\text{1 + x}^{4}}\text{dx}=\text{dt}$
$=\frac{\text{t}^{2}}{2}+\text{c}=\frac{1}{2}\Big[\tan^{-1}\text{(x}^{2})\Big]^{2}+\text{c}.$

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