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Question

$\int_{0}^{100}\left\{ \sqrt{x}\right\} \,dx$

Where $\{.\}$ denotes the fractional part of x.

My Approach

We know that $\left[x\right]$+$\left\{ x\right\} $=x $\Longrightarrow$$\left\{ x\right\} =x-\left[x\right]$

$\int_{0}^{100}\left\{ \sqrt{x}\right\} \,dx$ =$\int_{0}^{100}$$\sqrt{x}\,dx$ -$\int_{0}^{100}$$\left[\sqrt{x}\right]\,dx$=$\left[\frac{\sqrt{x^{3}}}{\frac{3}{2}}\right]_{0}^{100}-$$\int_{0}^{100}\left[\sqrt{x}\right]\,dx$.

I cannot solve $\int_{0}^{100}\left[\sqrt{x}\right]\,dx$. But I have an idea if somehow I can prove that $\left[\sqrt{x}\right]$ is a periodic function with period $p$ such that $p|100$ then $p\int_{0}^{\frac{100}{p}}\left[\sqrt{x}\right]\,dx$. That would be easy to solve.

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  • $\begingroup$ It is not a periodic function. Divide the range of integration into subintervals where $[\sqrt{x}]$ is constant $\endgroup$ – Abishanka Saha Dec 1 '17 at 7:11
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Guide:

\begin{align}\int_0^{100} \left[\sqrt{x} \right]\, dx &= \sum_{i=1}^{10} \int_{(i-1)^2}^{i^2} \left[\sqrt{x} \right]\,dx \\ &=\sum_{i=1}^{10}(i^2-(i-1)^2)\sqrt{(i-1)^2}\\ &= \sum_{i=1}^{10} ( 2i+1) (i-1)\end{align}

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  • $\begingroup$ Much better than my answer. $\endgroup$ – Piyush Divyanakar Dec 1 '17 at 7:15
  • $\begingroup$ Yours is more detailed. easier to understand perhaps. $\endgroup$ – Siong Thye Goh Dec 1 '17 at 7:15
  • $\begingroup$ I deal with the floor function rather than fractional function. Notice that if $x \in [(i-1)^2, i^2)$, then $[\sqrt{x}] = i-1$. $\endgroup$ – Siong Thye Goh Dec 1 '17 at 7:32
  • $\begingroup$ got it! Thanks! $\endgroup$ – Mohan Sharma Dec 1 '17 at 7:37
  • $\begingroup$ And you can get a closed form for a general perfect square upper bound by using en.wikipedia.org/wiki/Faulhaber%27s_formula $\endgroup$ – orion Dec 1 '17 at 8:45
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$$\{\sqrt x\}=\sqrt x - [\sqrt x]$$

Now see when $1 \le x < 4$. The integer part $[\sqrt x]=1$ Similarly $$1 \le x < 4 \implies [\sqrt x]=1\\ 4 \le x < 9 \implies [\sqrt x]=2 \\ 9 \le x < 16 \implies [\sqrt x]=3 \\ ... \\ 81 \le x < 100 \implies [\sqrt x]=9\\$$ So you have to do the following integral $$\int_0^{100}\{\sqrt x\}=\int_0^{100}\sqrt xdx-\left(\int_1^41dx+\int_4^92dx+...+\int_{81}^{100}9dx \right)$$

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