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

Question

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

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Answer

$\text{For I}=\int\frac{\text{x + 2}}{\text{2x}^{2}+\text{6x + 5}}\text{dx}=\frac{1}{4}\int\frac{\text{4x + 6 + 2}}{\text{2x}^{2}+\text{6x + 5}}\text{dx}$
$=\frac{1}{4}\int\frac{\text{4x + 6}}{\text{2x}^{2}+\text{6x}+\text{5}}\text{dx}+\frac{1}{4}\int\frac{\text{dx}}{\text{x}^{2}+\text{3x}+\frac{5}{2}}$

$=\frac{1}{4}\cdot\log|\text{2x}^{2}+\text{6x}+5|+\frac{1}{4}\int\frac{\text{dx}}{\Big(\text{x}+\frac{3}{2}\Big)^{2}+\Big(\frac{5}{2}-\frac{9}{4}\Big)}$

$=\frac{1}{4}\log|\text{2x}^{2}+\text{6x}+5|+\frac{1}{4}\int\frac{\text{dx}}{\Big(\text{x}+\frac{3}{2}\Big)^{2}+\Big(\frac{1}{2}\Big)^{2}}$

$=\frac{1}{4}\log|\text{2x}^{2}+\text{6x}+5|+\frac{1}{2}\tan^{-1}(\text{2x + 3})+\text{c}$.

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