Mathematics | MHT CET 2019 | Discussion Page

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

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

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

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

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

Option B

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

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

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

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

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

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

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

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