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Question Number 86246 by Rio Michael last updated on 27/Mar/20

$$\int \frac{x^2-1}{x^2+1}\frac{1}{\sqrt{1+x^4}}dx=?$$

Answered by TANMAY PANACEA. last updated on 27/Mar/20

$$\int \frac{1-\frac{1}{x^2}}{1+\frac{1}{x^2}}\times \frac{1}{x\sqrt{x^2+\frac{1}{x^2}}}dx$$$$\int \frac{d(x+\frac{1}{x})}{(x+\frac{1}{x})\sqrt{{(x+\frac{1}{x})}^2-2}}$$$$\int \frac{da}{a\sqrt{a^2-2}}$$$$a=\sqrt{2}sec\theta$$$$\int \frac{\sqrt{2}sec\theta tan\theta }{\sqrt{2}sec\theta \times \sqrt{2}tan\theta }d\theta$$$$\frac{1}{\sqrt{2}}{sec}^{-1}\left(\frac{a}{\sqrt{2}}\right)+c$$$$\frac{1}{\sqrt{2}}{sec}^{-1}\left(\frac{x+\frac{1}{x}}{\sqrt{2}}\right)+c$$$$$$

Commented by Rio Michael last updated on 27/Mar/20

$$\mathrm{thanks}\mathrm{sir}$$

Commented by TANMAY PANACEA. last updated on 27/Mar/20

$$mostwelcome$$

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