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) \[\log \left| \frac{x\sin x}{x+\cos x} \right|+c\]
E) \[\log \left| \frac{x}{x+\cos x} \right|+c\]
Correct Answer: E
Solution :
\[\frac{\cos x+x\sin x}{{{x}^{2}}+x\cos x}=\frac{(x+\cos x)-x+x\sin x}{x(x+\cos x)}\] \[=\frac{1}{x}-\frac{(1-\sin x)}{(x+\cos x)}\] \[\therefore \]\[\int{\frac{\cos x+x\sin x}{{{x}^{2}}+x\cos x}}dx\] \[\int{\left( \frac{1}{x}-\frac{1-\sin x}{x+\cos x} \right)}dx\] \[=\log x-\log (x+\cos x)+c\] \[=\log \left( \frac{x}{x+\cos x} \right)+c\]
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