EXAMPLE $3$ Find ${\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}{t}^2{e}^{6t}dt$

SOLUTION Notice that $t^2$ becomes simpler when differentiated $($whereas $e^{6t}$ is mostly unchanged when differentiated or integrated$),$ so we choose

$u=t^2$

$dv=e^{6t}dt$

Then $du=dt$

$v=$

Integration by parts gives

${\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}{t}^2{e}^{6t}dt=-\frac{1}{3}{\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}t{e}^{6t}dt$

The integral that we obtained, ${\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}t{e}^{6t}dt,$ is simpler than the original integral but is still not obvious. Therefore, we use integration by parts a second time, this time with $u=t$ and $dv=e^{6t}dt.$ Then $du=dt,$

Putting this in Equation $(3),$ we get

${\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}{t}^2{e}^{6t}dt=\frac{1}{6}{t}^2{e}^{6t}-$

${\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}t{e}^{6t}dt$