$\int \sqrt{x^{2} + 4 x + 1} dx$ is equal to

Easy

A

$(x + 2) \sqrt{x^{2} + 4 x + 1} - \frac{3}{2} \log |(x + 2) + \sqrt{x^{2} + 4 x + 1}| + C$
B

$\frac{(x + 2)}{2} \sqrt{x^{2} + 4 x + 1} + \frac{3}{2} \log |(x + 2) + \sqrt{x^{2} + 4 x + 1}| + C$
C

$\frac{(x + 2)}{2} \sqrt{x^{2} + 4 x + 1} - \frac{3}{2} \log |(x + 2) + \sqrt{x^{2} + 4 x + 1}| + C$
D

None of the above

Solution

Let
$\begin{array}{l} I = \int \sqrt{x^{2} + 4 x + 1} dx = \int \sqrt{x^{2} + 4 x + 1 - 2^{2} + 2^{2}} dx \\ = \int \sqrt{(x + 2)^{2} - (\sqrt{3})^{2}} dx \\ [∵ \int \sqrt{x^{2} - a^{2}} dx = \frac{x}{2} \sqrt{x^{2} - a^{2}} - \frac{a^{2}}{2} \log |x + \sqrt{x^{2} - a^{2}}| \end{array}$
$\Rightarrow I = \frac{x + 2}{2} \sqrt{x^{2} + 4 x + 1} - \frac{3}{2} \log |(x + 2) + \sqrt{x^{2} + 4 x + 1}| + C$

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