If [cdot] denotes the greatest integer functi | vedclass

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(D) Let $I = \int_{0}^{\pi} [\cos x] \, dx \quad \dots(1)$Using the property $\int_{0}^{a} f(x) \, dx = \int_{0}^{a} f(a-x) \, dx$,we get:$I = \int_{0

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$-\frac{\pi}{2}$

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(D) Let $I = \int_{0}^{\pi} [\cos x] \, dx \quad \dots(1)$Using the property $\int_{0}^{a} f(x) \, dx = \int_{0}^{a} f(a-x) \, dx$,we get:$I = \int_{0}^{\pi} [\cos(\pi - x)] \, dx = \int_{0}^{\pi} [-\cos x] \, dx \quad \dots(2)$Adding $(1)$ and $(2)$:$2I = \int_{0}^{\pi} ([\cos x] + [-\cos x]) \, dx$Since $[x] + [-x] = -1$ if $x 
otin \mathbb{Z}$ and $0$ if $x \in \mathbb{Z}$,and the set of points where $\cos x \in \mathbb{Z}$ in $[0, \pi]$ is finite (measure zero),we have:$2I = \int_{0}^{\pi} (-1) \, dx$$2I = -[x]_{0}^{\pi} = -\pi$$I = -\frac{\pi}{2}$

If $[\cdot]$ denotes the greatest integer function,then the integral $\int_{0}^{\pi} [\cos x] \, dx$ is equal to:

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