Question

For $0< x< \frac{\pi}{2}, \int\limits_{1 / 2}^{\sqrt{3} / 2} \ln \left(e^{\cos x}\right) \cdot d(\sin x)$ is equal to :

• A. $\frac{\pi}{12}$
• B. $\frac{\pi}{6}$
• C. $\frac{1}{4}[(\sqrt{3}-1)+(\sin \sqrt{3}-\sin 1)]$
• D. $\frac{1}{4}[(\sqrt{3}-1)-(\sin \sqrt{3}-\sin 1)]$

Answer 1

Answer: (A)

$I=\int\limits_{\pi / 6}^{\pi / 3} \cos x \cos x d x=\int\limits_{\pi / 6}^{\pi / 3} \cos ^2 x d x=\int\limits_{\pi / 6}^{\pi / 3} \sin ^2 x d x \Rightarrow 2 I=\int\limits_{\pi / 6}^{\pi / 3} d x=\frac{\pi}{6} \Rightarrow I=\frac{\pi}{12}$