Find: 3x+5 x^2+3x-18 dx. - Vidyadip

Question

Find: $\int\frac{3\text{x}+5}{\text{x}^2+3\text{x}-18}\text{ dx}.$

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Answer

$\text{l}\int\frac{3\text{x}+5}{\text{x}^2+3\text{x}-18}\text{ dx}$
$=\int\frac{\frac{3}{2}(2\text{x}+3)-\frac{9}{2}+5}{\text{x}^2+3\text{x}-18}\text{ dx}$
$=\frac{3}{2}\int\frac{2\text{x}+3}{\text{x}^2+3\text{x}-18}\text{dx}+\int\frac{\frac{1}{2}}{\text{x}^2+3\text{x}-18}$
$=\frac{3}{2}\int\frac{\frac{\text{d}}{\text{dx}}(\text{x}^2+3\text{x}-18)}{\text{x}^2+3\text{x}-18}\text{dx}+\frac{1}{2}\int\frac{\text{dx}}{\Big(\text{x}+\frac{3}{2}\Big)^2-18-\frac{9}{4}}$
$=\frac{3}{2}\text{ln}|\text{x}^2+3\text{x}-18|+\frac{1}{2}\int\frac{\text{dx}}{\Big(\text{x}+\frac{3}{2}\Big)^2-\big(\frac{9}{2}\big)^2}$
$=\frac{3}{2}\text{ln}|\text{x}^2+3\text{x}-18|+\frac{1}{18}\log\Big|\frac{\text{x}-3}{\text{x}+6}\Big|+\text{C}$

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