Question:
Evaluate $\int \frac{\log (1-x)}{x^{2}} d x$
Solution:
Use method of integration by parts
$y=\log (1-x) \int \frac{1}{x^{2}} d x-\int \frac{d}{d x} \log (1-x)\left(\int \frac{1}{x^{2}} d x\right) d x$
$y=-\log (1-x) \frac{1}{x}-\int \frac{1}{(1-x) x} d x$
$y=-\frac{1}{x} \log (1-x)-\int \frac{x+(1-x)}{(1-x) x} d x$
$y=-\frac{1}{x} \log (1-x)-\int \frac{1}{(1-x)}+\frac{1}{x} d x$
$y=-\frac{1}{x} \log (1-x)+\log (1-x)-\log x+c$
$y=\left(1-\frac{1}{x}\right) \log (1-x)-\log x+c$
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