5.5 The Substitution Rule/7

$\int x\sin {(x^{2})}dx$

${\begin{aligned}u&=x^{2}\\[2ex]du&=2xdx\\[2ex]{\frac {1}{2}}du&=dx\\[2ex]\end{aligned}}$

${\begin{aligned}\int x\sin {(x^{2})}dx&={\frac {1}{2}}\int \sin {(u)}du\\[2ex]&=-{\frac {1}{2}}\cos {(u)}+C\\[2ex]&=-{\frac {1}{2}}\cos {(x^{2})}+C\end{aligned}}$