There is one other tool that can sometimes be used to evaluate anti-derivatives that works when certain convergence conditions hold.

Suppose we know the anti-derivative of
$g(x,y)$
where
$g$
is some differentiable function of the parameter
$y$
, as well as a function of
$x$
. Then we can deduce that an anti-derivative of
$\frac{dg}{dy}$
is the derivative with respect to
$y$
of an anti-derivative of
$g$
.

For example, we know that an anti-derivative of
${e}^{xy}$
is
$\frac{{e}^{xy}}{y}$
. We may then deduce that an anti-derivative of
$\frac{d{e}^{xy}}{dy}$
is
$\frac{d\left({e}^{xy}/y\right)}{dy}$
.

You can take higher derivatives with respect to
$y$
here as well. This allows you to deduce a formula for an anti-derivative of a function of the form
${x}^{k}{e}^{ax}$
, by differentiating
$\frac{{e}^{xy}}{y}$$k$
times with respect to
$y$
and then setting
$y=a$
.

This method when it applies converts finding anti-derivatives to making appropriate differentiations. However, almost everything you can deduce this way can be gotten as well by integrating by parts judiciously.