Q. $\int_{}^{}\frac{\cos{x + x\sin x}}{x^{2} + x\cos x}\ dx$ is equal to
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) $\log{\left| \frac{x}{x + \cos x} \right| + c}$
Explanation
(d)
$\int_{}^{}\frac{\cos{x + x\sin x}}{x^{2} + x\cos x}\ dx = \int_{}^{}\frac{{x + cos}{x - x + x\sin x}}{x^{2} + x\cos x}\ dx$
$= \int_{}^{}\frac{dx}{x} - \int_{}^{}\frac{x(1 - \sin{x)}}{x(x + \cos{x)}}dx$
$= \log x - \log{(x + \cos{x)}} + c$
$= \log{\left| \frac{x}{x + \cos x} \right| + c}$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.