Let [x] and {x} be the integer part and fract | vedclass

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(B) Let $I = \int_0^5 [x]\{x\} dx$. Since $[x]$ is constant on intervals $[n, n+1)$,we can split the integral: $I = \sum_{n=0}^{4} \int_n^{n+1} [x]\{x

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$5$

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(B) Let $I = \int_0^5 [x]\{x\} dx$. Since $[x]$ is constant on intervals $[n, n+1)$,we can split the integral: $I = \sum_{n=0}^{4} \int_n^{n+1} [x]\{x\} dx$. For $x \in [n, n+1)$,$[x] = n$ and $\{x\} = x - n$. Thus,$I = \sum_{n=0}^{4} \int_n^{n+1} n(x-n) dx$. Let $t = x-n$,then $dt = dx$. When $x=n, t=0$ and when $x=n+1, t=1$. $I = \sum_{n=0}^{4} n \int_0^1 t dt = \sum_{n=0}^{4} n \left[ \frac{t^2}{2} \right]_0^1 = \sum_{n=0}^{4} n \left( \frac{1}{2} \right) = \frac{1}{2} (0+1+2+3+4) = \frac{10}{2} = 5$.

Let $[x]$ and $\{x\}$ be the integer part and fractional part of a real number $x$ respectively. The value of the integral $\int_0^5 [x]\{x\} dx$ is

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