Evaluate: (x - 3 ) x^ 2 + 3 x- 18 dx. - Vidyadip

Question

Evaluate: $\int(\text{x} - 3 )\sqrt{\text{x}^{2} + 3 \text{x}- 18 }\text{ dx}.$

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Answer

$\int(\text{x} - 3 ) \sqrt{\text{x}^{2} + 3 \text{x} - 18 }\text{ dx}$
$ = \frac{1}{2}\int(2 \text{x} + 3 ) \sqrt{\text{x}^{2} +3 \text{x} - 18}\text{ dx} - \frac{9}{2}\int\sqrt{\text{x}^{2} + 3 \text{x} - 18 |}\text{ dx}$
$ = \frac{1}{2}.\frac{2}{3}(\text{x}^{2} + 3 \text{x} - 18 )^{3/2} - \frac{9}{2}\int\sqrt{\bigg(\text{x} + 3/2\bigg)^{2} - \bigg(\frac{9}{2}\bigg)^{2}}\text{dx}$
$ =\frac{1}{3}(\text{x}^{2} + 3 \text{x} - 18 )^{3/2} -\frac{9}{2}$
$\frac{\bigg(\text{x} + \frac{3}{2}\bigg)}{2}\sqrt{\text{x}^{2} + 3\text{x}-18}-\frac{81}{8}\log\bigg|\text{x} + \frac{3}{2} + \sqrt{\text{x}^{2} + 3 \text{x}- 18 }\bigg| +\text{c}$
Or $ = \frac{1}{3}(\text{x}^{3} + 3\text{x} - 18)^{3/2} - \frac{9}{8}$
$(2 \text{x} + 3 ) \sqrt{\text{x}^{2} + 3\text{x} - 18 } -\frac{81}{2}\log\bigg|\text{x} + \frac{3}{2} + \sqrt{\text{x}^{2} + 3\text{x} - 18 }\bigg| + \text{c}.$

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