∫x2+3xdx is equal to

# $\int \sqrt{{\mathrm{x}}^{2}+3\mathrm{x}}\mathrm{dx}$ is equal to

1. A

$\frac{2\mathrm{x}+3}{2}\sqrt{{\mathrm{x}}^{2}+3\mathrm{x}}-\frac{9}{4}\mathrm{log}\left(\frac{2\mathrm{x}+3}{2}\right)+\sqrt{{\mathrm{x}}^{2}+3\mathrm{x}}\mid +\mathrm{C}$

2. B

$\frac{2\mathrm{x}+3}{4}\sqrt{{\mathrm{x}}^{2}+3\mathrm{x}}-\frac{9}{4}\mathrm{log}\left(\frac{2\mathrm{x}+3}{2}\right)+\sqrt{{\mathrm{x}}^{2}+3\mathrm{x}}\mid +\mathrm{C}$

3. C

$\left(\mathrm{x}+\frac{3}{2}\right)\sqrt{{\mathrm{x}}^{2}+3\mathrm{x}}-\frac{9}{8}\mathrm{log}\left(\mathrm{x}+\frac{3}{2}\right)+\sqrt{{\mathrm{x}}^{2}+3\mathrm{x}}\mid +\mathrm{C}$

4. D

None of the above

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### Solution:

Let

$\begin{array}{c}\mathrm{I}=\int \sqrt{{\mathrm{x}}^{2}+3\mathrm{x}}\mathrm{dx}=\int \sqrt{{\mathrm{x}}^{2}+3\mathrm{x}-{\left(\frac{3}{2}\right)}^{2}+{\left(\frac{3}{2}\right)}^{2}}\mathrm{dx}\\ =\int \sqrt{{\left(\mathrm{x}+\frac{3}{2}\right)}^{2}-{\left(\frac{3}{2}\right)}^{2}}\mathrm{dx}\\ \mathrm{I}=\frac{\mathrm{x}+\frac{3}{2}}{2}\sqrt{{\left(\mathrm{x}+\frac{3}{2}\right)}^{2}-{\left(\frac{3}{2}\right)}^{2}}-\frac{9}{4×2}\mathrm{log}\left(\left(\mathrm{x}+\frac{3}{2}\right)+\sqrt{{\mathrm{x}}^{2}+3\mathrm{x}}\mid +\mathrm{C}\end{array}$

$\begin{array}{r}\left[\because \int \sqrt{{\mathrm{x}}^{2}-{\mathrm{a}}^{2}}\mathrm{dx}=\frac{\mathrm{x}}{2}\sqrt{{\mathrm{x}}^{2}-{\mathrm{a}}^{2}}-\frac{{\mathrm{a}}^{2}}{2}\mathrm{log}\left|\mathrm{x}+\sqrt{{\mathrm{x}}^{2}-{\mathrm{a}}^{2}}\right|\right]\\ ⇒\mathrm{I}=\frac{2\mathrm{x}+3}{4}\sqrt{{\mathrm{x}}^{2}+3\mathrm{x}}-\frac{9}{8}\mathrm{log}\left|\left(\frac{2\mathrm{x}+3}{2}\right)+\sqrt{{\mathrm{x}}^{2}+3\mathrm{x}}\right|+\mathrm{C}\end{array}$

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