Integration by Parts II $($including tabular integration$)$

Recall from class that if you have an integral that looks solvable with integration by parts, but it also seems like it will need to be repeated many times over, you may be able to use tabular integration as a sort of shorthand for the work that goes into the problem.

you need:

$--$a consistent $"U"$ that can be repeatedly differentiated $($effectively all the way down to $0)$

$--$ a consistent $"dV"$ that can be repeatedly integrated

Every problem here is doable using integration by parts alone, but some may go faster if you use tabular integration $($note: $2$ of these problems can't really be done with the tabular method$).$

${\displaystyle \underset{0}{\overset{2}{}}}(2x+3)e^{2x}dx$

${\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}{x}^3{e}^{x^2}dx$

$($hint: $dV=x{e}^{x^2}dx)$

${\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}(x^4-x^2)cos(x)dx$

${\displaystyle \underset{0}{\overset{8}{}}}3x(1+x{)}^{-\frac{3}{2}}dx$

${\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}(4x^3-6x+1)ln(x)dx$