Mathematics

# The value of $\displaystyle \int_0^{\cfrac {\pi}{2}}\log \left (\frac {4+3 \sin x}{4+3 \cos x}\right )dx$ is

$0$

##### SOLUTION
Let $\displaystyle I=\int_0^{\frac {\pi}{2}}\log \left (\frac {4+3 \sin x}{4+3 \cos x}\right )dx$ .............. (1)
$\Rightarrow\displaystyle I=\int_0^{\frac {\pi}{2}}log \left [\frac {4+3\sin \left (\frac {\pi}{2}-x\right )}{4+3 \cos \left (\frac {\pi}{2}-x\right )}\right ]dx, \; (\because \int_0^a f(x) dx=\int_0^af(a-x)dx)$
$\Rightarrow\displaystyle I=\int_0^{\frac {\pi}{2}}\log \left (\frac {4+3 \cos x}{4+3 \sin x}\right )dx$ ........... (2)
Adding (1) and (2), we obtain
$2I\displaystyle =\int_0^{\frac {\pi}{2}}\left \{\log \left (\frac {4+3 \sin x}{4+3 \cos x}\right )+\log \left (\frac {4+3 \cos x}{4+3 \sin x}\right )\right \}dx$
$\Rightarrow\displaystyle 2I=\int_0^{\frac {\pi}{2}}\log \left (\frac {4+3 \sin x}{4+3 \cos x}\times \frac {4+3 \cos x}{4+3 \sin x}\right )dx$
$\Rightarrow 2I\displaystyle=\int_0^{\frac {\pi}{2}}\log 1 dx=\int_0^{\frac {\pi}{2}}0 dx$
$\Rightarrow I=0$
Hence, the correct Answer is C.

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Single Correct Medium Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 111

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