Question

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

$I=\underset{\pi /6}{\overset{\pi /3}{\int }}\cos x\cos xdx=\underset{\pi /6}{\overset{\pi /3}{\int }}{\cos }^{2}xdx=\underset{\pi /6}{\overset{\pi /3}{\int }}{\sin }^{2}xdx\Rightarrow 2I=\underset{\pi /6}{\overset{\pi /3}{\int }}dx=\frac{\pi }{6}\Rightarrow I=\frac{\pi }{12}$