∫x2(ax+b)−2dx is equal to

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

1. A

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

2. B

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

3. C

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

4. D

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

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### Solution:

$\mathrm{I}=\int \frac{{\mathrm{x}}^{2}}{\left(\mathrm{ax}+\mathrm{b}{\right)}^{2}}\mathrm{dx}$

put

$\begin{array}{r}=\frac{1}{{\mathrm{a}}^{3}}\left[\mathrm{ax}+\mathrm{b}-\frac{{\mathrm{b}}^{2}}{\mathrm{ax}+\mathrm{b}}-2\mathrm{blog}\left(\mathrm{ax}+\mathrm{b}\right)\right]+\mathrm{C}\\ =\frac{2}{{\mathrm{a}}^{2}}\left[\frac{\mathrm{x}}{2}-\frac{\mathrm{b}}{\mathrm{a}}\mathrm{log}\left(\mathrm{ax}+\mathrm{b}\right)\right]-\frac{{\mathrm{b}}^{2}}{{\mathrm{a}}^{3}\left(\mathrm{ax}+\mathrm{b}\right)}+\mathrm{C}\end{array}$

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