For 0 < x < 2 , _ 1 2 ^ 3 2 ln (e^ cos x ). d (sin , x) is equal … - Vidyadip

MCQ

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

✓
$\frac{\pi }{{12}}$
B
$\frac{\pi }{6}$
C
$\frac{1}{4}\,\,\left[ {\left( {\sqrt 3 \, - \,1} \right)\,\, + \,\,\left( {\sin \,\sqrt 3 \, - \,\sin \,1} \right)} \right]$
D
$\frac{1}{4}\,\,\left[ {\left( {\sqrt 3 \, - \,1} \right)\,\, - \,\,\left( {\sin \,\sqrt 3 \, - \,\sin \,1} \right)} \right]$

✓

Answer

Correct option: A.

$\frac{\pi }{{12}}$

a
$I = \int\limits_{\frac{\pi }{6}}^{\frac{\pi }{3}} \, \cos \,x \,cos \,x \,dx$

$=\int\limits_{\frac{\pi }{6}}^{\frac{\pi }{3}} cos^2 x\, dx$

$=\int\limits_{\frac{\pi }{6}}^{\frac{\pi }{3}} sin^2 x dx\, $

$\Rightarrow 2\, I = dx = \frac{\pi }{6}$

$ \Rightarrow I = \frac{\pi }{12}$

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