7.1 Integration By Parts/35

${\begin{aligned}&\int _{\sqrt {\frac {\pi }{2}}}^{\sqrt {\pi }}\ \theta ^{3}cos(\theta ^{2})d\theta &u=\theta ^{2}\\[2ex]&du=2\theta d\theta \\[2ex]&{\frac {1}{2}}du=\theta d\theta \end{aligned}}$

${\begin{aligned}&\int {}^{}u\cdot cos(u)du&w=u\\[2ex]&wu=du&dv=cos(u)dx&v=sin(u)\end{aligned}}$

${\begin{aligned}&u\cdot sin(u)-\int {}^{}sin(u)du&={\frac {1}{2}}u\cdot sin(u)+{\frac {1}{2}}cos(u){\bigg |}_{\frac {\pi }{2}}^{\pi }&={\frac {1}{2}}(\pi )\cdot sin(\pi )+{\frac {1}{2}}cos(\pi )-({\frac {1}{2}}({\frac {\pi }{2}})\cdot sin({\frac {\pi }{2}})+{\frac {1}{2}}cos({\frac {\pi }{2}}))\\[2ex]&={\frac {1}{2}}(\pi )(0)+{\frac {1}{2}}(-1)-({\frac {\pi }{4}}+{\frac {1}{2}}(0)\\[2ex]&=-{\frac {1}{2}}-{\frac {\pi }{4}}\end{aligned}}$