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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 d g d y is the derivative with respect to y of an anti-derivative of g .

For example, we know that an anti-derivative of e x y is e x y y . We may then deduce that an anti-derivative of d e x y d y is d ( e x y / y ) d y .

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 a x , by differentiating e x y 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.

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