Question

The value of the integral ${\int }_0^{\pi /4}\frac{\sin x+\cos x}{3+\sin 2x}dx$ is

• A. $\log 2$
• B. $\log 3$
• C. $\left(1/4\right)\log 3$
• D. $\left(1/8\right)\log 3$

Answer 1

Answer: (C)

$\left(1/4\right)\log 3$

Let $I={\int }_0^{\pi /4}\frac{\sin x+\cos x}{3+\sin 2x}dx={\int }_0^{\pi /4}\frac{\sin x+\cos x}{4-{(\sin x-\cos x)}^2}dx$
Substitute $\sin x-\cos x=t$
$I={\int }_{-1}^0\frac{dt}{4-t^2}=\frac{1}{4}\log 3$