∫ π 2 π θ 3 c o s ( θ 2 ) d θ u = θ 2 d u = 2 θ d θ 1 2 d u = θ d θ {\displaystyle {\begin{aligned}&\int _{\sqrt {\frac {\pi }{2}}}^{\sqrt {\pi }}\ \theta ^{3}cos(\theta ^{2})d\theta &u=\theta ^{2}\\[2ex]&du=2\theta d\theta \\[2ex]&{\frac {1}{2}}du=\theta d\theta \end{aligned}}}
∫ u ⋅ c o s ( u ) d u w = u w u = d u d v = c o s ( u ) d x v = s i n ( u ) {\displaystyle {\begin{aligned}&\int {}^{}u\cdot cos(u)du&w=u\\[2ex]&wu=du&dv=cos(u)dx&v=sin(u)\end{aligned}}}
u ⋅ s i n ( u ) − ∫ s i n ( u ) d u = 1 2 u ⋅ s i n ( u ) + 1 2 c o s ( u ) | π 2 π = 1 2 ( π ) ⋅ s i n ( π ) + 1 2 c o s ( π ) − ( 1 2 ( π 2 ) ⋅ s i n ( π 2 ) + 1 2 c o s ( π 2 ) ) = 1 2 ( π ) ( 0 ) + 1 2 ( − 1 ) − ( π 4 + 1 2 ( 0 ) = − 1 2 − π 4 {\displaystyle {\begin{aligned}&u\cdot sin(u)-\int {}^{}sin(u)du&={\frac {1}{2}}u\cdot sin(u)+{\frac {1}{2}}cos(u){\bigg |}_{\frac {\pi }{2}}^{\pi }&={\frac {1}{2}}(\pi )\cdot sin(\pi )+{\frac {1}{2}}cos(\pi )-({\frac {1}{2}}({\frac {\pi }{2}})\cdot sin({\frac {\pi }{2}})+{\frac {1}{2}}cos({\frac {\pi }{2}}))\\[2ex]&={\frac {1}{2}}(\pi )(0)+{\frac {1}{2}}(-1)-({\frac {\pi }{4}}+{\frac {1}{2}}(0)\\[2ex]&=-{\frac {1}{2}}-{\frac {\pi }{4}}\end{aligned}}}
![{\displaystyle {\begin{aligned}&\int _{\sqrt {\frac {\pi }{2}}}^{\sqrt {\pi }}\ \theta ^{3}cos(\theta ^{2})d\theta &u=\theta ^{2}\\[2ex]&du=2\theta d\theta \\[2ex]&{\frac {1}{2}}du=\theta d\theta \end{aligned}}}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/d8112ae7d8c6fa9c2c271010a2ed8c56983761c3)
![{\displaystyle {\begin{aligned}&\int {}^{}u\cdot cos(u)du&w=u\\[2ex]&wu=du&dv=cos(u)dx&v=sin(u)\end{aligned}}}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/fd6c08bdf9b292b425379bdd565409a54e686ed1)
![{\displaystyle {\begin{aligned}&u\cdot sin(u)-\int {}^{}sin(u)du&={\frac {1}{2}}u\cdot sin(u)+{\frac {1}{2}}cos(u){\bigg |}_{\frac {\pi }{2}}^{\pi }&={\frac {1}{2}}(\pi )\cdot sin(\pi )+{\frac {1}{2}}cos(\pi )-({\frac {1}{2}}({\frac {\pi }{2}})\cdot sin({\frac {\pi }{2}})+{\frac {1}{2}}cos({\frac {\pi }{2}}))\\[2ex]&={\frac {1}{2}}(\pi )(0)+{\frac {1}{2}}(-1)-({\frac {\pi }{4}}+{\frac {1}{2}}(0)\\[2ex]&=-{\frac {1}{2}}-{\frac {\pi }{4}}\end{aligned}}}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/ff563caf2e84b9c5f3ab3bb46bb0ca4cc4019802)