Evaluate the definite integral int_2^5 2[x] , | vedclass

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(A) We are given the integral $I = \int_2^5 2[x] \, dx$. Since $[x]$ is the greatest integer function,it takes constant integer values in intervals of

Vedclass · Accepted Answer

$18$

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(A) We are given the integral $I = \int_2^5 2[x] \, dx$. Since $[x]$ is the greatest integer function,it takes constant integer values in intervals of length $1$. We split the integral at the integers $3$ and $4$: $I = 2 \left( \int_2^3 [x] \, dx + \int_3^4 [x] \, dx + \int_4^5 [x] \, dx ight)$ For $x \in [2, 3)$,$[x] = 2$. For $x \in [3, 4)$,$[x] = 3$. For $x \in [4, 5)$,$[x] = 4$. Substituting these values: $I = 2 \left( \int_2^3 2 \, dx + \int_3^4 3 \, dx + \int_4^5 4 \, dx ight)$ $I = 2 \left( [2x]_2^3 + [3x]_3^4 + [4x]_4^5 ight)$ $I = 2 \left( (6 - 4) + (12 - 9) + (20 - 16) ight)$ $I = 2 \left( 2 + 3 + 4 ight) = 2 	imes 9 = 18$.

Evaluate the definite integral $\int_2^5 2[x] \, dx$,where $[x]$ denotes the greatest integer function $\leq x$.

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