First slide

Methods of integration

Question

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

Easy

A

$\frac{{(1 - x)}^{2 - n}}{2 - n} + \frac{{(1 - x)}^{1 - n}}{1 - n} + c$
B

$\frac{{(1 - x)}^{2 + n}}{2 + n} - \frac{{(1 - x)}^{1 + n}}{2 + n} + c$
C

$\frac{{(1 - x)}^{2 - n}}{2 - n} - \frac{{(1 - x)}^{1 - n}}{1 - n} + c$
D

$\frac{{(1 - x)}^{2 - n}}{2 + n} + \frac{{(1 - x)}^{1 - n}}{1 + n} + c$

Solution

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

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