Find: (2x + 5) 10 - 4x - 3x^ 2 dx - Vidyadip

Question

Find: $\int (2x + 5) \sqrt{10 - 4x - 3x^{2}} \text{ dx}$

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Answer

$\text{I} = \int {(\text{2x + 5)}} \sqrt{10 - \text{4x - 3x}^{2}} \text{dx}$
$= -\frac{1}{3} \int(-4-6\text{x}) \sqrt{10 - \text{4x - 3x}^{2}} \text{dx} + \frac{11}{3} \int \sqrt{10 - \text{4x - 3x}^{2}} \text{ dx}$
$= -\frac{2}{9} (10 - \text{4x - 3x}^{2})^{3/2} + \frac{11\sqrt{3}}{3} \int \sqrt{\bigg(\frac{\sqrt{34}}{3}\bigg)^{2} - \bigg(\text{x} - \frac{2}{3}\bigg)^{2}} \text{dx}$
$= -\frac{2}{9} (10 - \text{4x - 3x}^{2})^{3/2} + \frac{11\sqrt{3}}{3}\Bigg[\frac{\big(\text{x} - \frac{2}{3}\big) - \sqrt{\big(\frac{\sqrt{34}}{3}\big)^{2} - \big(\text{x} - \frac{2}{3}\big)^{2}}}{2} + \frac{17}{9} \sin^{-1} \frac{3\text{x - 2}}{\sqrt{34}} \Bigg] + \text{C}$

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