B I Feel Weird Using Integral Tables

[Thinkaholic]

Sorry if this is in the wrong place. Sometimes I do really stupid things on integrals (use a method that gets me nowhere, make a mistake while factoring quickly, etc.) I have always been reluctant on using tables because I always felt stupid using them. I feel like I have to reinvent every general integral out there as I do one. I know, I am stupid for even saying that. Being stupid is my thing.
I also feel that if I use a table often, I’ll forget how to do simple integrals.
Anyways, do you guys use integral tables often? Is using an integral table a sign that someone has no understanding of integration techniques whatsoever? I know these are dumb questions so I’m sorry.

 

[FactChecker]

You did not say if these are calculus class problems or real-world integrals. In calc class, you should learn how to do those yourself because that is what the class is for. But in practice, very few integrals can be derived in a closed form and algorithms must be used. And even many of the closed form integrals are related to special functions that are very difficult to figure out. So don't feel bad about using tables on those.

That being said, in physics there is a lot of use of standard sets of orthogonal basis functions where the inner product is an integral and you should know the properties of those.

 

[jedishrfu]

We stand on the shoulders of giants and those giants stand on a great tortoise. Don't be that tortoise, stand on the shoulders of your favorite giant and enjoy the view.

Many integrals are real works of art that only if you know the trick will you be able to cleanly integrate it. Sure, it will make you stronger to integrate as many as you can meanwhile the Earth moves onward in its orbit and you should be learning new stuff.

Here’s one strategy I learned recently which at first seems totally out in left field:

https://en.m.wikipedia.org/wiki/Integration_using_parametric_derivatives

 

[FactChecker]

We stand on the shoulders of giants and those giants stand on a great tortoise. Don't be that tortoise, stand on the shoulders of your favorite giant and enjoy the view.

Ha! I like that. That's the "Quote of the day."

 

[Thinkaholic]

I am talking about hard integrals that sometimes pop up on these forums. I can do high school integrals with ease. I definitely understand the properties of integrals, the different methods, and how most integrals on the tables were derived. Honestly, I don’t even know why I wrote this [emoji23]. Oops

 

[FactChecker]

I am talking about hard integrals that sometimes pop up on these forums. I can do high school integrals with ease. I definitely understand the properties of integrals, the different methods, and how most integrals on the tables were derived. Honestly, I don’t even know why I wrote this [emoji23]. Oops

It was a good question. Many people who have finished calc classes but have not done a lot in real applications think that they should be able to calculate all integrals in a closed-form solution. Nothing could be further from the truth. They usually don't tell you that in calc classes.

 

[jedishrfu]

I am talking about hard integrals that sometimes pop up on these forums. I can do high school integrals with ease. I definitely understand the properties of integrals, the different methods, and how most integrals on the tables were derived. Honestly, I don’t even know why I wrote this [emoji23]. Oops

You wrote it because it bothers you that you can't do them and that's understandable as we can't either. We learn several techniques to compute them using integration by parts or some trig identity and then we're done. The magic behind each of the other harder integrals could well be someone's PhD thesis in math. It may come out of some investigation into a new area and wall-lah a new integral has been discovered.

You also wrote it so I got could come up with a funny quote but that's an example of quantum entanglement and beyond the scope of this thread.

 

[mathwonk]

I never used integral tables. I felt it would stunt my growth in learning how to do them. But i never had a job where I needed to do some goofy integral I wasn't really interested in doing. I was a college math prof for 40 years, and always had to relearn to do certain specific integrals every time I taught the course. I always felt that I should keep trying to do things over and over until I finally understood the "why" so I could remember them. Eventually I got pretty good, to where once I did an integral that my copy of Mathematica got stuck on. I may have typed it in wrong though because at a later date the program did it easily, or maybe they jazzed up the software. So basically do not use tables if you are trying to learn to do integrals since that does not help, but do use them if you just need to know an answer and move on, that's my opinion.

 

[PAllen]

I never used integral tables. I felt it would stunt my growth in learning how to do them. But i never had a job where I needed to do some goofy integral I wasn't really interested in doing. I was a college math prof for 40 years, and always had to relearn to do certain specific integrals every time I taught the course. I always felt that I should keep trying to do things over and over until I finally understood the "why" so I could remember them. Eventually I got pretty good, to where once I did an integral that my copy of Mathematica got stuck on. I may have typed it in wrong though because at a later date the program did it easily, or maybe they jazzed up the software. So basically do not use tables if you are trying to learn to do integrals since that does not help, but do use them if you just need to know an answer and move on, that's my opinion.

Mathematica may have changed since then, but I had an experience in which I communicated with their tech support. I had an integral mathematica couldn't do, but which I felt should be exactly integrable, though quite messy. So I found a substitution that simplified it a bit, and mathematica easily solved it from there. To me, that seemed like a bug. Mathematica told me that they don't actually do substitutions, - they have several very general integration approaches, then simplify. Thus, they claimed it was not a bug but a design limitation.

 

[jedishrfu]

Sometimes the software approach will use recursion which can get bogged down if it goes too deep as in out of heap space or stack space.

 

[FactChecker]

I only care about getting the job done. There are literally thousands of definite integrals in the book of tables for which I have no interest in knowing how they were done. Tricks don't interest me unless they give me a more profound insight into something.

 

[mathwonk]

the topic of exactly which intgrals are finite combinations of elementary functions does interest me though, i.e. exactly which integrals are "doable" at least in theory. There are some nice articles out there on this. Of course in this subject, as in almost all subjects, the kicker is the fact that most polynomials are not actually factorable into irreducibles in practice, although theoretically they are.

 

[George Jones]

Mathematica may have changed since then, but I had an experience in which I communicated with their tech support. I had an integral mathematica couldn't do, but which I felt should be exactly integrable, though quite messy. So I found a substitution that simplified it a bit, and mathematica easily solved it from there. To me, that seemed like a bug. Mathematica told me that they don't actually do substitutions, - they have several very general integration approaches, then simplify. Thus, they claimed it was not a bug but a design limitation.

I have had similar experiences with Maple.

Because of stuff like this, it is important to know standard elementary techniques like substitution, integration by parts, partial fractions, etc., as well as some "tricks". It might take a combination of these techniques to massage an integral into a form that is in a table, or that can be performed by standard software.