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(C) The given integral is $I = \int_{\frac{3 \pi}{4}}^\pi \left[ \sin x + \left[ \frac{4 x}{\pi} \right] \right] dx$.Since $[a + n] = [a] + n$ for any

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$3 \pi / 4$

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(C) The given integral is $I = \int_{\frac{3 \pi}{4}}^\pi \left[ \sin x + \left[ \frac{4 x}{\pi} \right] \right] dx$.Since $[a + n] = [a] + n$ for any integer $n$,we have $\left[ \sin x + \left[ \frac{4 x}{\pi} \right] \right] = [\sin x] + \left[ \frac{4 x}{\pi} \right]$.For the interval $\frac{3 \pi}{4} \leq x \leq \pi$,the value of $\sin x$ lies between $0$ and $\frac{1}{\sqrt{2}}$.Since $0 \leq \sin x Now,for $\frac{3 \pi}{4} \leq x Therefore,$\left[ \frac{4 x}{\pi} \right] = 3$.Substituting these values into the integral,we get $I = \int_{\frac{3 \pi}{4}}^\pi (0 + 3) dx$.$I = 3 \int_{\frac{3 \pi}{4}}^\pi dx = 3 [x]_{\frac{3 \pi}{4}}^\pi = 3 \left( \pi - \frac{3 \pi}{4} \right) = 3 \left( \frac{\pi}{4} \right) = \frac{3 \pi}{4}$.

If $[.]$ represents the greatest integer function,then evaluate the integral: $\int_{\frac{3 \pi}{4}}^\pi \left[ \sin x + \left[ \frac{4 x}{\pi} \right] \right] dx$.

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