A) \[\log [(1+\sin x)(2+\sin x)]+c\]
B) \[\log \frac{2+\sin x}{1+\sin x}+c\]
C) \[\log \frac{1+\sin x}{2+\sin x}+c\]
D) None of these
Correct Answer: C
Solution :
Put \[\sin x=t\Rightarrow \cos x\,dx=dt,\] then \[\int_{{}}^{{}}{\frac{\cos x}{(1+\sin x)(2+\sin x)}}\,dx=\int_{{}}^{{}}{\frac{dt}{(t+1)(t+2)}}\] \[=\int_{{}}^{{}}{\frac{1}{t+1}dt-\int_{{}}^{{}}{\frac{1}{t+2}dt}}=\log \left( \frac{t+1}{t+2} \right)+c=\log \left( \frac{\sin x+1}{\sin x+2} \right)+c\].
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