19.6 Differentiation with Respect to a Parameter

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.