Question

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

• A. $\log \left|\frac{x+\cos x}{x}\right|+c$
• B. $\log |x(x+\cos x)|+c$
• C. $\log \left|\frac{x}{x+\cos x}\right|+c$
• D. $\log \lfloor x(\cos x+x\sin x)|+c$

Answer 1

Answer: (C)

$\log \left|\frac{x}{x+\cos x}\right|+c$

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

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

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

$=\int \left(\frac{1}{x}\right)-\left[\frac{1-\sin x}{x+\cos x}\right]dx$

$=\int \left(\frac{1}{x}\right)dx-\int \left[\frac{f^{\prime }\left(x\right)}{f\left(x\right)}\right]dx$

$=f\left(x\right)=x+\cos x$

$=ln\left|x\right|-ln\left|f\left(x\right)\right|+C$

$=ln\left|x\right|-ln|x+\cos x|+C$

$=ln\left|\frac{x}{x+\cos x}\right|+C$

Hence, solved.