Hello All, I had put in the back of my Hilbert paper a little bit about a method called Iterated Integration by Parts. My then Calc prof, Dr. Boal, taught this trick to me years ago. I've never seen it in any Calc book - and I have quite a few. I was wondering if any of you have seen this method before and if so, was it just shown to you or was it in a book? Just curious. I'm surprised it is not featured in Calc books under the integration by parts section. Clay
Slightly OT: Iterated Integration by Parts
Started by ●March 4, 2005
Reply by ●March 4, 20052005-03-04
Clay wrote:> Hello All, > > I had put in the back of my Hilbert paper a little bit about a method > called Iterated Integration by Parts. My then Calc prof, Dr. Boal, > taught this trick to me years ago. I've never seen it in any Calc book > - and I have quite a few. I was wondering if any of you have seen this > method before and if so, was it just shown to you or was it in a book? > Just curious. I'm surprised it is not featured in Calc books under the > integration by parts section. > > ClayThere's the freshman-calculus integration of exp(ax)sin(bx)dx. Integration by parts leads to exp(ax) a ( - ------ cos(bx) + - | exp(ax)cos(bx)dx, which looks like a step b b ) backward. Integrating exp(ax)cos(bx)dx by parts and collecting terms finally leads to a solution that some familiar with Laplace transforms will recognize. Jerry -- Engineering is the art of making what you want from things you can get. �����������������������������������������������������������������������
Reply by ●March 4, 20052005-03-04
Jerry Avins wrote:>> > There's the freshman-calculus integration of exp(ax)sin(bx)dx. > Integration by parts leads to > > exp(ax) a ( > - ------ cos(bx) + - | exp(ax)cos(bx)dx, which looks like a step > b b ) > > backward. Integrating exp(ax)cos(bx)dx by parts and collecting terms > finally leads to a solution that some familiar with Laplacetransforms> will recognize. > > Jerry > -- > Engineering is the art of making what you want from things you canget.>=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF= =AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF= =AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF Jerry, The above is certainly true. What I'm refering to as iterated integration by parts is the simple tabular method for doing repeated integration by parts. For sure, the method is based on doing a simple integration by parts, but its beauty lies in the simple way of keeping up with the intermediate terms and sign changes for repeating the process. Almost all of my Calc books will show the integration by parts rule and then assign some problems where the rule has to be applied more than once. But what I've never seen in a book is the tabular method. Amazingly the book "Integration Handbook" which is a real treasure chest of integration techniques doesn't even have this one. Dr. Boal, who taught iterated integration by parts to me, was a graduate of MIT. Maybe they have kept this trick a secret there ;-) He told me he learned it while in school. I know MIT is famous for its Integration Bee they hold every year. What other tricks are being held up their sleeves? Clay
Reply by ●March 4, 20052005-03-04
"Clay" <physics@bellsouth.net> wrote in message news:1109962928.178508.153440@g14g2000cwa.googlegroups.com... Jerry Avins wrote:>> > There's the freshman-calculus integration of exp(ax)sin(bx)dx. > Integration by parts leads to > > exp(ax) a ( > - ------ cos(bx) + - | exp(ax)cos(bx)dx, which looks like a step > b b ) > > backward. Integrating exp(ax)cos(bx)dx by parts and collecting terms > finally leads to a solution that some familiar with Laplacetransforms> will recognize. > > Jerry > -- > Engineering is the art of making what you want from things you canget.>����������������������������������������������������������������������� Jerry, The above is certainly true. What I'm refering to as iterated integration by parts is the simple tabular method for doing repeated integration by parts. For sure, the method is based on doing a simple integration by parts, but its beauty lies in the simple way of keeping up with the intermediate terms and sign changes for repeating the process. Almost all of my Calc books will show the integration by parts rule and then assign some problems where the rule has to be applied more than once. But what I've never seen in a book is the tabular method. Amazingly the book "Integration Handbook" which is a real treasure chest of integration techniques doesn't even have this one. Dr. Boal, who taught iterated integration by parts to me, was a graduate of MIT. Maybe they have kept this trick a secret there ;-) He told me he learned it while in school. I know MIT is famous for its Integration Bee they hold every year. What other tricks are being held up their sleeves? Clay Hi Clay - http://students.uwsp.edu/jwhit216/By%20Parts.htm and http://marauder.millersville.edu/~bikenaga/calculus/parts/partspf.html seem to be examples , is this what you are talking about? Incidentally , I remember seeing a note a while back about the merits of doing it the wrong way sometimes. If I remember,the example was integrating e^(-x) by parts to get a polynomial expansion with increasingly tight bounds on the accuracy of the answer , providied x<1 ( but it was a while ago so I could easilly have forgotten something or just be plain wrong again). best of luck - Mike
Reply by ●March 4, 20052005-03-04
Hello Mike, Thanks - that's it. I'm glad it is at least out on the web. I'm just suprised it hasn't made it to the books yet. I've also seen where I. by P. has been used to grind out a poly. Cool idea. Thanks for the link. Clay
Reply by ●March 4, 20052005-03-04
"Clay" <physics@bellsouth.net> wrote in message news:1109966632.395972.39900@g14g2000cwa.googlegroups.com...> Thanks - that's it. I'm glad it is at least out on the web. I'm just > suprised it hasn't made it to the books yet.I have always heard of this referred to as 'tabular integration' and google gets a lot of hits on it. I actually was informed of the technique by a Norwegian undergraduate who I was teaching. I think he said he was taught it in high school or something. It is in some calc books, although sometimes just as a problem. -- write(*,*) transfer((/17.392111325966148d0,6.5794487871554595D-85, & 6.0134700243160014d-154/),(/'x'/)); end
