7.1 Integration By Parts/7

${\begin{aligned}&=\int x^{2}sin(\pi x)\\[2ex]&U=x^{2},\\[1ex]du=2xdx,\\[1ex]dv=sin(\pi x)dx,\\[1ex]v=-{\frac {1}{\pi }}cos(\pi x)\\[2ex]&=-{\frac {x^{2}}{\pi }}cos(\pi x)+{\frac {2}{\pi }}\int xcos(\pi x)dx\\[2ex]&U=x,\\[1ex]du=dx,\\[1ex]dv=cos(\pi x)dx,\\[1ex]v={\frac {1}{\pi }}sin(\pi x)\\[2ex]&=-{\frac {x^{2}}{\pi }}cos(\pi x)+{\frac {2}{\pi }}[{\frac {x}{\pi }}sin(\pi x)-{\frac {1}{\pi }}\int sin(\pi x)dx]\\[2ex]&=-{\frac {x^{2}}{\pi }}cos(\pi x)+{\frac {2}{\pi }}[{\frac {x}{\pi }}sin(\pi x)-{\frac {1}{\pi ^{2}}}cos(\pi x)]+c\\[2ex]\end{aligned}}$