$\int \sqrt{2 - 3 x - 4 x^{2}} d x$

Moderate

A

$\frac{1}{16} (8 x + 3) \sqrt{2 - 3 x - 4 x^{2}} + \frac{41}{64} \sin^{- 1} (\frac{8 x + 3}{\sqrt{41}}) + C$
B

$\frac{1}{16} (8 x + 3) \sqrt{2 - 3 x - 4 x^{2}} - \frac{41}{64} \sin^{- 1} (\frac{8 x + 3}{\sqrt{41}}) + C$
C

$\frac{1}{16} (8 x + 3) \sqrt{2 - 3 x - 4 x^{2}} + \frac{41}{64} \cos^{- 1} (\frac{8 x + 3}{\sqrt{41}}) + C$
D

$\frac{1}{16} (8 x + 3) \sqrt{2 - 3 x - 4 x^{2}} - \frac{41}{64} \cos^{- 1} (\frac{8 x + 3}{\sqrt{41}}) + C$

Solution

$\begin{array}{l} 2 \int \sqrt{{(\frac{\sqrt{41}}{8})}^{2} - {(x + \frac{3}{8})}^{2}} d x \\ Using \int \sqrt{a^{2} - x^{2}} d x = \frac{x}{2} \sqrt{a^{2} - x^{2}} + \frac{a^{2}}{2} \sin^{- 1} (\frac{x}{a}) + C \\ = \frac{1}{16} (8 x + 3) \sqrt{2 - 3 x - 4 x^{2}} + \frac{41}{64} \sin^{- 1} (\frac{8 x + 3}{\sqrt{41}}) + C \end{array}$

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