Evaluate: 1 + x^ 2 1 + x^ 4 dx - Vidyadip

Question

Evaluate:$\int \frac{1 + x^{2}}{1 + x^{4}} \text{dx}$

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Answer

$\text{I} \int \frac{1 + x^{2}}{1 + x^{4}} \text{dx} = \int \frac{1 +\frac{1}{x^{2}}}{x^{2} + \frac{1}{x^{2}}}\text{dx} = \int \frac{1 +\frac{1}{x^{2}}}{\bigg( x - \frac{1}{x}\bigg)^{2} + \bigg(\sqrt{2}\bigg)^{2}} \text{dx}$$= \frac{1}{\sqrt{2}} \tan^{-1} \bigg(x-\frac{\frac{1}{x}}{\sqrt{2}}\bigg) + c $
$= \frac{1}{\sqrt{2}} \tan^{-1}\bigg(\frac{x^{2} -1}{\sqrt{2x}}\bigg) + c $

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