Evaluate: x + 2 x^ 2 + 5x + 6 dx.

Question

Evaluate: $\int\frac{\text{x} + 2}{\sqrt{\text{x}^{2} + 5\text{x} + 6 }}\text{dx}.$

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Answer

$\text{I} = \int\frac{\text{x} + 2 }{\sqrt{\text{x}^{2} + 5\text{x} + 6 }}\text{dx} = \int\frac{\frac{1}{2}(2\text{x} + 5 ) - \frac{1}{2}}{\sqrt{\text{x}^{2} + 5 \text{x} + 6 }}\text{dx}$
$ =\frac{1}{2}\int\frac{2\text{x} + 5 }{\sqrt{\text{x}^{2} + 5\text{x} + 6 }}\text{dx}-\frac{1}{2}\int\frac{\text{dx}}{\sqrt{\big(\text{x} + 5/2\big)^{2} - \bigg(\frac{1}{2}\bigg)^{2}}}$
$ = \sqrt{\text{x}^{2} + 5\text{x} + 6 } -\frac{1}{2}\log\bigg|\bigg(\text{x} + \frac{5}{2}\bigg) + \sqrt{\text{x}^{2} + 5\text{x} + 6 }\bigg| + \text{c}.$

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