Home  18.013A  Chapter 19 


There is one other tool that can sometimes be used to evaluate antiderivatives that works when certain convergence conditions hold.
Suppose we know the antiderivative 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 antiderivative of is the derivative with respect to y of an antiderivative of g.
For example, we know that an antiderivative of We may then deduce that an antiderivative of
You can take higher derivatives with respect to y here as well. This allows you to deduce a formula for an antiderivative of a function of the form x^{k} e^{ax}, by differentiating k times with respect to y and then setting y = a.
This method when it applies converts finding antiderivatives to making appropriate differentiations. However, almost everything you can deduce this way can be gotten as well by integrating by parts judiciously.
