DeGuerre wrote:I think that a lot of the trouble is that there is usually some fiddle factor that you need to get exactly right. In this example:
∫ x e^(2x) dx
You would say that e^(2x) gets no more complicated when you integrate, and I'd agree with you because we use the same notion of "more complicated. For a beginning calculus student, however, 1/2 e^(2x) sure as hell looks more complicated because of the fiddle factor.Farabor wrote:When I teach it, especially for beginning calculus problems, there are pretty much exactly two cases you'll use it. In both cases, its freaking obvious what to do.
Nothing personal, but this sounds like cargo cult mathematics to me. If I gave your students a problem like ∫ 2 x e^(x^2) dx, they'd probably be lost even though it's an almost trivially easy problem.Farabor wrote:Case 2 (The annoying one). There are two terms, one is cosine/sine, and the other is an exponential. Then you have to do the double integration by parts/combine the integral method.
Clearly students would need to be taught that (1/2)*e^2x and e*2x are basically the same complexity (if it isn't obvious to them) but this would happen much earlier in calculus anyway.
since in ∫ 2 x e^(x^2) dx the e^x^2 term doesn't get easier when you integrate or differentiate students would quite happily put integration by parts away and try something that might work (like substitution) or if they're bright they'll try differentiating e^x^2 an realise something seems familiar. This method of identifying questions where you would use int. by parts seems to work fine in this case.
DeGuerre wrote:And this is a perfect example where a Risch-like method makes far more sense.
what makes far more sense to you might not for everyone else. Your method looks intimidating to me so I imagine it looks intimidating to most students. When I was taught this it was sort of left to our intuition when you used integration by parts twice and combine the integral. If this (exponential and trig) is the only example of this in the scope of the course then teaching it as the one more difficult case is likely how you'd go about teaching it and I imagine I'd have found it easier.