$\int x^{2} (ax + b)^{- 2} dx$ is equal to

Moderate

A

$\frac{2}{a^{2}} \{\frac{x}{2} - \frac{b}{a} \log (ax + b)\} + C$
B

$\frac{2}{a^{2}} \{\frac{x}{2} - \frac{b}{a} \log (ax + b)\} - \frac{b^{2}}{a^{3} (ax + b)} + C$
C

$\frac{2}{a^{2}} \{\frac{x}{2} + \frac{b}{a} \log (ax + b)\} + \frac{b^{2}}{a^{3} (ax + b)} + C$
D

$\frac{2}{a^{2}} \{\frac{x}{2} + \frac{b}{a} \log (ax + b)\} - \frac{b^{2}}{a^{3} (ax + b)} + C$

Solution

$I = \int \frac{x^{2}}{(ax + b)^{2}} dx$
put $ax + b = t \Rightarrow dx = \frac{1}{a} dt and x = (\frac{t - b}{a})$
$\begin{aligned} ∴ I = \frac{1}{a^{3}} \int \frac{(t - b)^{2}}{t^{2}} dt = \frac{1}{a^{3}} \int (1 + \frac{b^{2}}{t^{2}} - \frac{2 b}{t}) dt \\ = \frac{1}{a^{3}} (t - \frac{b^{2}}{t} - 2 blog t) + C \end{aligned}$
$\begin{aligned} = \frac{1}{a^{3}} [ax + b - \frac{b^{2}}{ax + b} - 2 blog (ax + b)] + C \\ = \frac{2}{a^{2}} [\frac{x}{2} - \frac{b}{a} \log (ax + b)] - \frac{b^{2}}{a^{3} (ax + b)} + C \end{aligned}$

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