5.5 The Substitution Rule/7

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

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

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