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Question Number 196459 by RoseAli last updated on 25/Aug/23

Answered by Frix last updated on 25/Aug/23

$$\mathrm{Use}{\mathrm{Ostrogradski}}^{\prime }\mathrm{s}\mathrm{Method}\mathrm{to}\mathrm{get}$$$$\int \frac{x^4+x^2+1}{{(x^2+1)}^3}dx=\frac{x(2x^2+3)}{2(x^2+1)}-\frac{1}{2}\int \frac{dx}{x^2+1}=$$$$=\frac{x(2x^2+3)}{2(x^2+1)}-\frac{{\tan }^{-1}x}{2}+C$$$$\Rightarrow$$$$\mathrm{For}$$$$\sqrt{2}-1⩽x⩽\sqrt{2}+1$$$$\mathrm{the}\mathrm{value}\mathrm{of}\mathrm{the}\mathrm{integral}\mathrm{is}$$$$2-\frac{\pi }{8}$$

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