1 x 2 - x 3 ? d x =

Question

$\int \frac{1}{x^{2} - x^{3}} d x =$

Smart Achievers · Accepted Answer

(4)

Ans. $\log$ $| \frac{x}{1 - x} |$ $- \frac{1}{x} + C$

Explanation :

$\int \frac{d x}{x^{2} - x^{3}} = \int \frac{1 - x + x}{x^{2} (1 - x)} d x$

$= \int \frac{(1 - x) d x}{x^{2} (1 - x)} + \int \frac{x d x}{x^{2} (1 - x)}$

$= \int \frac{1}{x^{2}} d x + \int \frac{d x}{x (1 - x)}$

$= - \frac{1}{x} + \int \frac{d x}{x} + \int \frac{d x}{1 - x}$

$= - \frac{1}{x} + \log$ $| x |$ $- \log$ $| 1 - x |$ $+ C$ $(∵ - 1 < x < 1)$

$= \log$ $| \frac{x}{1 - x} |$ $- \frac{1}{x} + C$

$= \log (\frac{x}{1 - x}) + C$

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