Question

The value of $\int \frac{\cos x+x\sin x}{x^2+x\cos x}dx$ is

• A. $\log \left|\frac{\sin x}{1+\cos x}\right|+C$
• B. $\log \left|\frac{\sin x}{x+\cos x}\right|+C$
• C. $\log \left|\frac{2\sin x}{x+\cos x}\right|+C$
• D. $\log \left|\frac{x}{x+\cos x}\right|+C$

Answer 1

The correct option is D $\log \left|\frac{x}{x+\cos x}\right|+C$

$\int \frac{\cos x+x\sin x}{x^2+x\cos x}dx$
$=\int \frac{x+\cos x-x+x\sin x}{x^2+x\cos x}dx$
$=\int \frac{dx}{x}-\int \frac{x(1-\sin x)}{x(x+\cos x)}dx$
$=\log x-\log (x+\cos x)+C$
$=\log \left|\frac{x}{x+\cos x}\right|+C$