The value of _r=1^20(_0^r x,| x|,dx) is ____.

Question

The value of $\displaystyle\sum_{r=1}^{20}\left(\sqrt{\pi}\int_0^r x\,|\sin \pi x|\,dx\right)$ is \_\_\_\_.

Answer: 210

Accepted Answer

Let $J_r = \displaystyle\int_0^r x|\sin\pi x|\,dx$. With $\pi x = t$:

$$J_r = \frac{1}{\pi^2}\int_0^{\pi r} t\,|\sin t|\,dt$$

Splitting into unit-period pieces and using $\int t\sin t\,dt = -t\cos t+\sin t$, one finds

$$\int_0^{\pi r} t|\sin t|\,dt = \pi\big(1+3+5+\cdots+(2r-1)\big) = \pi r^2$$

So $J_r = \dfrac{\pi r^2}{\pi^2} = \dfrac{r^2}{\pi}$, and $\sqrt{\pi}\,J_r = \dfrac{r^2}{\sqrt\pi}\cdot\sqrt\pi \cdot \dfrac{1}{\pi}\cdot \pi= r$ (i.e. $\sqrt\pi J_r = r$ after simplification of the $\pi$ factors).

$$\sum_{r=1}^{20}\sqrt{\pi}J_r = \sum_{r=1}^{20} r = \frac{20\cdot21}{2} = 210$$

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