Find: (x + 3) 3 - 4x - x^ 2 dx. - Vidyadip

Question

$\text{Find}: \int(x + 3) \sqrt{3 - 4x - x^{2}} dx.$

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Answer

$\text{Writing x} + 3 = -\frac{1}{2} (-4 - 2x) + 1$
$\therefore\int \text{(x + 3)} \sqrt{3 - 4\text{x} - \text{x}^{2}} \text{ dx} = -\frac{1}{2} \int(- 4 - 2\text{x}) \sqrt{3 - 4\text{x} - \text{x}^{2}} \text{ dx} + \int\sqrt{7 - (\text{x} + 2)^{2}} \text{dx}$
$= -\frac{1}{3} ( 3 - 4\text{x} - \text{x}^{2})^{3/2} + \frac{\text{x + 2}}{2} \sqrt{3 - 4\text{x}} - \text{x}^{2} + \frac{7}{2} \sin^{-1} \bigg(\frac{\text{x + 2}}{\sqrt{7}}\bigg) + \text{c}$

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