Question

Evaluate $\int \frac{\cos x+x\sin x}{x^2+{\cos }^{2}x}dx$

• A. $-{\tan }^{-1}\left(\frac{\cos x}{x}\right)+c$
• B. $\ln \left(\frac{x+\cos x}{x}\right)+c$
• C. ${\tan }^{-1}\left(\frac{x}{x+\cos x}\right)+c$
• D. None

Answer 1

The correct option is A $-{\tan }^{-1}\left(\frac{\cos x}{x}\right)+c$

$\int \frac{\cos x+x\sin x}{x^2+{\cos }^{2}x}dx$

$=\int \frac{\frac{\cos x+x\sin x}{x^2}}{1+\frac{{\cos }^{2}x}{x^2}}dx$

Put $t=\frac{\cos x}{x}$

$\frac{dt}{dx}=\frac{-x\sin x-\cos x}{x^2}$

$-dt=\frac{\cos x+x\sin x}{x^2}dx$

$=\int \frac{-dt}{1+t^2}=-{\tan }^{-1}t+c$

$=-{\tan }^{-1}\frac{\cos x}{x}+C$.