5.5 The Substitution Rule/3

$\int x^{2}{\sqrt {x^{3}+1}}dx$

${\begin{aligned}u&=x^{3}+1\\[2ex]du&=3x^{2}dx\end{aligned}}$

$\int {\frac {1}{3}}du{\sqrt {u}}$

${\begin{aligned}{\frac {1}{3}}\int du(u^{\frac {1}{2}})\\[2ex]={\frac {1}{2}}u^{\frac {3}{2}}\end{aligned}}$

${\frac {1}{2}}(x^{3}+1)^{\frac {3}{2}}+C$