Evaluate: 5x + 3 x^ 2 +4x + 10 . - Vidyadip

Question

Evaluate: $\int\frac{\text{5x + 3}}{\sqrt{\text{x}^{2}+\text{4x + 10}}}$.

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Answer

$\text{I}=\int\frac{\text{5x + 3}}{\sqrt{\text{x}^{2}+\text{4x + 10}}}\text{dx}=\int\frac{\frac{5}{2}\text{(2x + 4)}-7}{\sqrt{\text{x}^{2}+\text{4x}+10}}$$=\frac{5}{2} \int\frac{\text{2x + 4}}{\sqrt{\text{x}^{2}+\text{4x + 10}}}\text{dx}-7 \int\frac{1}{\sqrt{\text{(x+2)}^{2}+(\sqrt{6})^{2}}}\text{dx}$
$ =5\cdot\sqrt{\text{x}^{2}+\text{4x + 10}}-7\log|\text{(x + 2)}+\sqrt{\text{x}^{2}+\text{4x}+10}|+\text{c}$.

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