Question

Given ${\int }_0^{\frac{\pi }{2}}\frac{dx}{1+\sin x+\cos x}=\log 2$. Then the value of the definite integral ${\int }_0^{\frac{\pi }{2}}\frac{\sin x}{1+\sin x+\cos x}dx$ is equal to?

• A. $\frac{1}{2}\log 2$
• B. $\frac{\pi }{2}-\log 2$
• C. $\frac{\pi }{4}-\frac{1}{2}\log 2$
• D. $\frac{\pi }{2}+\log 2$

Answer 1

The correct option is C $\frac{\pi }{4}-\frac{1}{2}\log 2$

$I={\int }_0^{\frac{\pi }{2}}\frac{\sin x}{1+\sin x+\cos x}dx$
$={\int }_0^{\frac{\pi }{2}}\frac{\cos xdx}{1+\sin x+\cos x}$
or $2I={\int }_0^{\frac{\pi }{2}}\frac{\sin x+\cos x+1-1}{\sin x+\cos x+1}dx$
$=\frac{\pi }{2}-\log 2$
or $I=\frac{\pi }{4}-\frac{1}{2}\log 2$