Evaluate x-x x x(x+ x) d x. - Vidyadip

Question

Evaluate $\int \frac{\sin x-x \cos x}{x(x+\sin x)} d x$.

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Answer

Let
$
\begin{aligned}
I & =\int \frac{\sin x-x \cos x}{x(x+\sin x)} d x \\
& =\int \frac{(x+\sin x)-x(1+\cos x)}{x(x+\sin x)} d x \\
& =\int \frac{x+\sin x}{x(x+\sin x)} d x-\int \frac{x(1+\cos x)}{x(x+\sin x)} d x \\
& =\int \frac{1}{x} d x-\int \frac{1+\cos x}{x+\sin x} d x \\
& =\log |x|-\log |x+\sin x|+\text { C Ans. }
\end{aligned}
$

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