Integral Calculus Question 98
Question: Given $ \int\limits_0^{\pi /2}{\frac{dx}{1+\sin x+\cos x}=\log ,2.} $ Then the value of the definite integral $ \int\limits_0^{\pi /2}{\frac{\sin x}{1+\sin x+\cos x}dx} $ is equal to
Options:
A) $ \frac{1}{2}\log 2 $
B) $ \frac{\pi }{2}-\log 2 $
C) $ \frac{\pi }{4}-\frac{1}{2}\log 2 $
D) $ \frac{\pi }{2}+\log 2 $
Show Answer
Answer:
Correct Answer: C
Solution:
[c] $ I=\int\limits_0^{\pi /2}{\frac{\sin xdx}{1+\sin x+\cos x}} $ $ =\int\limits_0^{\pi /2}{\frac{\cos xdx}{1+\sin x+\cos x}} $ Or $ 2I=\int\limits_0^{\pi /2}{\frac{\sin x+\cos x+1-1}{\sin x+\cos x+1}dx} $ $ 2I=\frac{\pi }{2}-\log 2 $ Or $ I=\frac{\pi }{4}-\frac{1}{2}\log 2 $
BETA
