Question

The value of the integral $\int\limits_{\pi/6}^{\pi/2}\left(\frac{1+\sin\, 2x+ \cos\, 2x }{\sin \, x +\cos x}\right)dx$ is equal to

• A. 16
• B. 8
• C. 4
• D. 1

Answer 1

Answer: (D)

Let $I=\int\limits_{\pi / 6}^{\pi / 2}\left(\frac{1+\sin 2 x+\cos 2 x}{\sin x+\cos x}\right) d x$
$=\int\limits_{\pi / 6}^{\pi / 2}\left(\frac{1+2 \sin x \cos x+2 \cos ^{2} x-1}{(\sin x+\cos x)}\right) d x$
$=\int\limits_{\pi / 6}^{\pi / 2} \frac{2 \cos x(\sin x+\cos x)}{(\sin x+\cos x)} d x$
$=\int\limits_{\pi / 6}^{\pi / 2} 2 \cos x d x=2[\sin x]_{\pi / 6}^{\pi / 2}$
$=2\left(\sin \frac{\pi}{2}- \sin \frac{\pi}{6}\right)$
$=2\left(1-\frac{1}{2}\right)=2 \times \frac{1}{2}=1$