Integral Calculus Question 38
Question: $ \int_{{}}^{{}}{\frac{\cos x}{(1+\sin x)(2+\sin x)}\ dx=} $
[Roorkee 1979]
Options:
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
Show Answer
Answer:
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 ( \frac{t+1}{t+2} )+c=\log ( \frac{\sin x+1}{\sin x+2} )+c $ .
BETA
