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Integration by Substitution

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Question

simplify $\int (\frac{x}{a + b x}) d x =$

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Solution

$\frac{x}{a + b x} = \frac{1}{b} \times \frac{a + b x - a}{a + b x} = \frac{1}{b} - \frac{a}{b (a + b x)}$
$\int \frac{x d x}{a + b x} = \frac{1}{b} \int d x - \frac{a}{b} \int \frac{d x}{a + b x} = \frac{x}{b} - \frac{a}{b^{2}} ln (a + b x) + c$

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