Evaluate the following integrals: 1 3x^2+5x+7 dx

Question

Evaluate the following integrals:

$\int\frac{1}{\sqrt{3\text{x}^2+5\text{x}+7}}\text{ dx}$

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Answer

$\int\frac{\text{dx}}{\sqrt{3\text{x}^2+5\text{x}+7}}$
$=\int\frac{\text{dx}}{\sqrt{3\big(\text{x}^2+\frac{5}{3}\text{x}+\frac{7}{3}}\big)}$
$=\frac{1}{\sqrt3}\int\frac{\text{dx}}{\sqrt{\text{x}^2+\frac{5}{3}\text{x}+\big(\frac{5}{6}\big)^2-\big(\frac{5}{6}\big)^2+\frac{7}{3}}}$
$=\frac{1}{\sqrt3}\int\frac{\text{dx}}{\sqrt{\big(\text{x}+\frac{5}{6}\big)^2-\frac{25}{36}+\frac{7}{3}}}$
$=\frac{1}{\sqrt3}\int\frac{\text{dx}}{\sqrt{\big(\text{x}+\frac{5}{6}\big)^2+\frac{-25+84}{36}}}$
$=\frac{1}{\sqrt3}\int\frac{\text{dx}}{\sqrt{\big(\text{x}+\frac{5}{6}\big)^2+\frac{59}{36}}}$
$=\frac{1}{\sqrt3}\int\frac{\text{dx}}{\sqrt{\big(\text{x}+\frac{5}{6}\big)^2+\Big(\sqrt{\frac{59}{36}}\Big)^2}}$
$=\frac{1}{\sqrt3}\log\Bigg|\text{x}+\frac{5}{6}+\sqrt{\big(\text{x}+\frac{5}{6}\big)^2+\frac{59}{36}}\Bigg|+\text{C}$
$=\frac{1}{\sqrt3}\log\Bigg|\text{x}+\frac{5}{6}+\sqrt{\text{x}^2+\frac{5}{3}\text{x}+\frac{7}{3}}\Bigg|+\text{C}$

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