Discussion Forum



1)

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


A) $\log |\frac{x\sin x}{x^{}+ \cos x}|+c$

B) $\log |\frac{x}{x^{}+ \cos x}|+c$

C) $\log |\cos x+x\sin x|+c$

D) $\log |x^{2}+x \cos x|+c$

Answer:

Option B

Explanation:

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

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

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

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

=$\log|x|-\int\frac{1-\sin x}{x+\cos x}dx$

=$\log|x|-\log(x+\cos x)+c$

 = $\log |\frac{x}{x^{}+ \cos x}|+c$