TMUA Yotta Paper 1 · Question 20

The function $F(x)$ , where $x$ is a non-negative real number, is the result of subtracting the integer part of $x$ from $x$ . For example, $F(3)=0$ , $F(5.43)=0.43$ , $F(\pi)=0.14159265\ldots$ . Find an expression for:

\int_0^{\sqrt{k}}F(x^2)\,\dd x

where $k$ is a positive integer.

\frac13 k^{3/2}-F(\frac13 k^{3/2})

\frac13 k^{3/2}-\sum_{n=0}^k n(\sqrt{n+1}-\sqrt n)

\frac13 k^{3/2}+\sum_{n=0}^{k-1}n(\sqrt{n+1}-\sqrt n)

\frac13 k^{3/2}-\sum_{n=0}^{k-1}n(\sqrt{n+1}+\sqrt n)

\frac13 k^{3/2}+\left(\sum_{n=0}^{k-1}\sqrt n\right)+(k-1)\sqrt k

\frac13 k^{3/2}+\left(\sum_{n=0}^{k-1}\sqrt n\right)+(1-k)\sqrt k

examday

Yotta Paper 1 · Q20