5.5 The Substitution Rule/21

$\int {\frac {\cos {({\sqrt {t}})}}{\sqrt {t}}}dt$

${\begin{aligned}u&={\sqrt {u}}\\[2ex]du&={\frac {1}{2}}\ {\frac {1}{\sqrt {t}}}dx\\[2ex]2du&={\frac {1}{\sqrt {t}}}dx\end{aligned}}$

${\begin{aligned}\int {\frac {\cos {({\sqrt {t}})}}{\sqrt {t}}}dt&=2\int \cos {u}du\\[2ex]&=2\sin {u}+c\\[2ex]&=2\sin({\sqrt {u}})+c\\[2ex]\end{aligned}}$