Find 2x (x^ 2 + 1)(x^ 2 + 2)^ 2 dx. - Vidyadip

Question

Find $\int \frac{2x}{(x^{2} + 1)(x^{2} + 2)^{2}} \text{d}x.$

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Answer

$\int \frac{\text{2x}}{(\text{x}^{2} + 1)\text{(x}^{2} + 2)^{2}} = \int \frac{\text{dy}}{\text{(y + 1) (y + 2)}^{2}}$ [by substituting x² = y]
$ = \int \frac{\text{dy}}{\text{y + 1}} - \int \frac{\text{dy}}{\text{y + 2}} - \int\frac{\text{dy}}{(\text{y + 2)}^{2}}$ (using partial fraction)
$= \log \text{(y + 1)} - \log (\text{y + 2)} + \frac{1}{\text{y + 2}} + \text{C}$
$= \log (\text{x}^{2} + 1) - \log \text{(x}^{2} + 2) + \frac{1}{\text{x}^{2} + 2} + \text{C}$

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