Question

Evaluate $\int \frac{x^2}{(x^2+2)({2x}^2+1)}dx$

Answer 1

$\int \frac{x^2}{(x^2+2)(2x^2+1)}dx$

Apply integration by parts.

$=\int \frac{2}{3(x^2+2)}-\frac{1}{3(2x^2+1)}dx$

$=\int \frac{2}{3(x^2+2)}dx-\int \frac{1}{3(2x^2+1)}dx$

$=\frac{\sqrt{2}}{3}{\tan }^{-1}\left(\frac{x}{\sqrt{2}}\right)-\frac{1}{3\sqrt{2}}{\tan }^{-1}\left(\sqrt{2}x\right)$

$=\frac{\sqrt{2}}{3}{\tan }^{-1}\left(\frac{x}{\sqrt{2}}\right)-\frac{1}{3\sqrt{2}}{\tan }^{-1}\left(\sqrt{2}x\right)+C$