= ∫ x 2 s i n ( π x ) U = x 2 , d u = 2 x d x , d v = s i n ( π x ) d x , v = − 1 π c o s ( π x ) = − x 2 π c o s ( π x ) + 2 π ∫ x c o s ( π x ) d x U = x , d u = d x , d v = c o s ( π x ) d x , v = 1 π s i n ( π x ) = − x 2 π c o s ( π x ) + 2 π [ x π s i n ( π x ) − 1 π ∫ s i n ( π x ) d x ] = − x 2 π c o s ( π x ) + 2 π [ x π s i n ( π x ) − 1 π 2 c o s ( π x ) ] + c {\displaystyle {\begin{aligned}&=\int x^{2}sin(\pi x)\\[2ex]U=x^{2},\\[2ex]du=2xdx,\\[2ex]dv=sin(\pi x)dx,\\[2ex]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,\\[2ex]du=dx,\\[2ex]dv=cos(\pi x)dx,\\[2ex]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}}}
![{\displaystyle {\begin{aligned}&=\int x^{2}sin(\pi x)\\[2ex]U=x^{2},\\[2ex]du=2xdx,\\[2ex]dv=sin(\pi x)dx,\\[2ex]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,\\[2ex]du=dx,\\[2ex]dv=cos(\pi x)dx,\\[2ex]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}}}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/e1f4c78ca5da7692fcb2588e330870d4798985a4)