∫ 0 π ( t s i n 3 t ) d t = ∫ 0 π 1 2 t 2 1 2 s i n 3 t 2 = ∫ 0 π 1 2 π 2 1 2 s i n 3 π 2 = π 3 {\displaystyle {\begin{aligned}\int _{0}^{\pi }(tsin3t)dt\\[2ex]&=\int _{0}^{\pi }{\frac {1}{2}}t^{2}{\frac {1}{2}}sin3t^{2}\\[2ex]&=\int _{0}^{\pi }{\frac {1}{2}}\pi ^{2}{\frac {1}{2}}sin3\pi ^{2}\\[2ex]&={\frac {\pi }{3}}\end{aligned}}}
![{\displaystyle {\begin{aligned}\int _{0}^{\pi }(tsin3t)dt\\[2ex]&=\int _{0}^{\pi }{\frac {1}{2}}t^{2}{\frac {1}{2}}sin3t^{2}\\[2ex]&=\int _{0}^{\pi }{\frac {1}{2}}\pi ^{2}{\frac {1}{2}}sin3\pi ^{2}\\[2ex]&={\frac {\pi }{3}}\end{aligned}}}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/797c64f017c703d9b53672c37ce1d48c9bc67b90)