Problem

Find

Solution

This function is the product of x², which is easily differentiable, and e^x, which is easily integrable. So integration by parts, with u = x² and dv = e^x dx, gives

This reduces the problem of finding the original integral to the simpler problem of finding

This problem is solved for us in Example 1 in Section 4.5:

So all together,

What if we had chosen something different for u and dv? Say we let u = e^x, dv = x² dx. Then we have du = e^x dx,

and

This equation is true, but it doesn't simplify the problem, since integrating x³e^x is as hard to solve as the original problem.