자연계열 2-2번

주어진 식에 의하여 구간 $\displaystyle [2026,2028]$에서 $\displaystyle f(x)=2^{1013}(x-2026)\sin\pi(x-2026)$이다.

절댓값 함수의 부호가 바뀌는 점은 $\displaystyle \sin\pi x=\frac12$이므로, $\displaystyle x=\frac16$, $\displaystyle \frac56$이다.따라서, $\displaystyle S=2^{1012}\left(\int_0^{\frac16}(x-2x\sin\pi x)dx+\int_{\frac56}^1(x-2x\sin\pi x)dx+\int_{\frac16}^{\frac56}(2x\sin\pi x-x)dx\right)$ …(*)

부정적분 $\displaystyle \int x\sin\pi x\,dx=-\frac1\pi x\cos\pi x+\frac1{\pi^2}\sin\pi x+C$이고, 치환적분에 의해$\displaystyle \int_{\frac56}^1(x-2x\sin\pi x)dx=\int_0^{\frac16}(1-x)-2(1-x)\sin\pi x\,dx$이므로,$\displaystyle \begin{aligned}S&=2^{1012}\left(\int_0^{\frac16}1-2\sin\pi x\,dx+\int_{\frac16}^{\frac56}2x\sin\pi x-x\,dx\right)\\&=2^{1012}\left\{\Biggl[x+\frac2\pi\cos\pi x\Biggr]_0^{\frac16}+\Biggl[-\frac2\pi x\cos\pi x+\frac2{\pi^2}\sin\pi x-\frac{x^2}2\Biggr]_{\frac16}^{\frac56}\right\}\\&=2^{1012}\left(\frac16+\frac{\sqrt3}\pi-\frac2\pi+\frac{\sqrt3}\pi-\frac13\right)=2^{1012}\left(-\frac16-\frac2\pi+\frac{2\sqrt3}\pi\right)\end{aligned}$이다.따라서, $\displaystyle p=-\frac16$, $\displaystyle q=-2$, $\displaystyle r=2$이다.