Evaluate the definite integral int_0^2 [2x] , | vedclass

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(B) To evaluate $\int_0^2 [2x] \, dx$,we break the integral at the points where $2x$ is an integer,i.e.,$2x = 0, 1, 2, 3, 4$. This corresponds to $x =

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$3$

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(B) To evaluate $\int_0^2 [2x] \, dx$,we break the integral at the points where $2x$ is an integer,i.e.,$2x = 0, 1, 2, 3, 4$. This corresponds to $x = 0, 1/2, 1, 3/2, 2$.The integral becomes:$\int_0^{1/2} 0 \, dx + \int_{1/2}^1 1 \, dx + \int_1^{3/2} 2 \, dx + \int_{3/2}^2 3 \, dx$$= 0 \cdot (1/2 - 0) + 1 \cdot (1 - 1/2) + 2 \cdot (3/2 - 1) + 3 \cdot (2 - 3/2)$$= 0 + 1/2 + 2(1/2) + 3(1/2)$$= 0 + 0.5 + 1 + 1.5 = 3$.

Evaluate the definite integral $\int_0^2 [2x] \, dx$,where $[.]$ denotes the greatest integer function.

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