$\int \sqrt{x^{2} + 3 x} d x$ is equal to
(where $C$ is integration constant)

\frac{2 x + 3}{4} \sqrt{x^{2} + 3 x} - \frac{9}{8} log ∣ ∣ ∣ x + \frac{3}{2} + \sqrt{x^{2} + 3 x} ∣ ∣ ∣ + C

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\frac{x}{2} \sqrt{x^{2} + 3 x} - \frac{9}{8} log ∣ ∣ ∣ x + \frac{3}{2} + \sqrt{x^{2} + 3 x} ∣ ∣ ∣ + C

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\frac{x}{2} \sqrt{x^{2} + 3 x} - \frac{9}{8} {sin}^{- 1} (\frac{2 x + 3}{2}) + C

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\frac{2 x + 3}{4} \sqrt{x^{2} + 3 x} - \frac{9}{8} {sin}^{- 1} (\frac{2 x + 3}{2}) + C

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Solution

The correct option is A $\frac{2 x + 3}{4} \sqrt{x^{2} + 3 x} - \frac{9}{8} log ∣ ∣ ∣ x + \frac{3}{2} + \sqrt{x^{2} + 3 x} ∣ ∣ ∣ + C$
$I = \int \sqrt{x^{2} + 3 x} d x = \int \sqrt{x^{2} + 3 x + \frac{9}{4} - \frac{9}{4}} d x = \int \sqrt{{(x + \frac{3}{2})}^{3} - {(\frac{3}{2})}^{2}} d x$

We know that,
$\int \sqrt{x^{2} - a^{2}} d x = \frac{x}{2} \sqrt{x^{2} - a^{2}} - \frac{a^{2}}{2} log | x + \sqrt{x^{2} - a^{2}} | + C ∴ I = \frac{(x + \frac{3}{2})}{2} \sqrt{x^{2} + 3 x} - \frac{\frac{9}{4}}{2} log ∣ ∣ ∣ (x + \frac{3}{2}) + \sqrt{x^{2} + 3 x} ∣ ∣ ∣ + C = \frac{(2 x + 3)}{4} \sqrt{x^{2} + 3 x} - \frac{9}{8} log ∣ ∣ ∣ (x + \frac{3}{2}) + \sqrt{x^{2} + 3 x} ∣ ∣ ∣ + C$

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