Find ${\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}{t}^2{e}^{5t}dt$

Solution

Notice that $t^2{e}^{5t}is$ mostly unchanged when differentiated $or$ integratedu$=t^2$ and $dv=e^{5t}dt.$

Then $du=(,)dt$ and $v=$

Integration $by$ parts gives

${\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}{t}^2{e}^{5t}dt=(,)-\frac{2}{5}{\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}t{e}^{5t}dt$