Evaluate: x (1 - x) ( 2 - cos x) dx

Question

Evaluate: $\int \frac{\sin x}{(1 - \cos x) ( 2 - cos x)} \text{dx}$

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Answer

$\text{I} =\int \frac{\sin x}{(1 - \cos x) ( 2 - cos x)} \text{dx}$$\text{Let} - \cos \text{x} = \text{t} \Rightarrow \sin \text{x dx} = \text{dt}$
$\therefore \text{I}= \int \frac{dt}{(1+ t) ( 2 + t)} = \int\frac{dt}{1 + t} - \int \frac{dt}{2 + t}$
$= \log | 1 + t | - \log| 2 + t| + c$
$= \log | 1 - \cos x | - \log | 2 - \cos x | + c$
OR
$\log \bigg|\frac{1 - cos x}{2 - cos x}\bigg| + c$

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