Question

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

Answer 1

$I=\int \frac{\cos x+x\sin x}{x(x+\cos x)}dx$

$I=\int \frac{(\cos x+x)+(x\sin x-x)}{x(x+\cos x)}dx$

$I=\int \frac{1}{x}\frac{(x+\cos x)}{(x+\cos x)}+\int \frac{x(\sin x-1)}{x(x+\cos x)}dx$

$I=\int \frac{1}{x}dx+\int \frac{(\sin x-1)}{x+\cos x}dx$

Let $x+\cos x=u$

$du=(1-\sin x)dx$

$I=ln|x|+\int \frac{-(1-\sin x)}{u}\frac{du}{(1-\sin x)}$

$I=\ln |x|-\ln |u|+c$

$I=\ln |x|-\ln |x+\cos x|+c$.