Question

Solve: $\int \frac{\sin x-x\cos x}{x(x+\sin x)}dx$

Answer 1

$\int \frac{\sin x-x\cos x}{x(x+\sin x)}dx=\int \frac{(x+\sin x)-x(1+\cos x)}{x(x+\sin x)}dx=\int [\frac{1}{x}-\frac{1+\cos x}{x+\sin x}]dx=\int \frac{1}{x}dx-\int \frac{1+\cos x}{x+\sin x}dxPutx+\sin x=z\Rightarrow (1+\cos x)dx=dz=\ln |x|-\int \frac{dz}{z}=\ln |x|-\ln |z|+C=\ln |x|-\ln |x+\sin x|+C=\ln \left|\frac{x}{x+\sin x}\right|+C$