Calculus Where can I find an extensive table of integrals?

[gulfcoastfella]

I'm reading through an undergrad physics book, and the author says he looked up the answer to the below integral in a table. I've tried to find tables of integrals with this integral included in them, but have failed so far. Can someone direct me to an exhaustive table of integrals and their solutions? The integral in question follows:

$$ \int_{-1}^{1} {\frac{\arcsin(x)}{(1+ax)^2}} dx , \quad 0<a<1 $$

 

[Dr. Courtney]

http://www.lepp.cornell.edu/~ib38/tmp/reading/Table_of_Integrals_Series_and_Products_Tablicy_Integralov_Summ_Rjadov_I_Proizvedennij_Engl._2.pdf

 

[gulfcoastfella]

Thank you Dr. Courtney, that's a great resource and I'll keep it bookmarked.

 

[Imager]

http://www.lepp.cornell.edu/~ib38/t...ralov_Summ_Rjadov_I_Proizvedennij_Engl._2.pdf

The link isn't working for me...

 

[Demystifier]

Not really a table of integrals, but can be used in that way (and in many other ways):
http://www.wolframalpha.com/

 

[jack476]

Does the author say anywhere in the book which table he used? The physics textbooks I've read tend to refer readers to which tables they were using cite their sources rather extensively.

 

[bcrowell]

Tables of integrals have been made obsolete by computer software. Maxima is free and open-source.

maxima -r "assume(a>0); assume(a<1); tex(integrate(asin(x)/(1+a*x)^2,x,-1,1));"

result: $-{{\pi\,\sqrt{1-a^2}+\pi\,a-\pi}\over{2\,a^3-2\,a}}-{{\pi\,\sqrt{1- a^2}-\pi\,a-\pi}\over{2\,a^3-2\,a}}$

 

[DevacDave]

Tables of integrals have been made obsolete by computer software. Maxima is free and open-source.

maxima -r "assume(a>0); assume(a<1); tex(integrate(asin(x)/(1+a*x)^2,x,-1,1));"

result: $-{{\pi\,\sqrt{1-a^2}+\pi\,a-\pi}\over{2\,a^3-2\,a}}-{{\pi\,\sqrt{1- a^2}-\pi\,a-\pi}\over{2\,a^3-2\,a}}$

True and great point about Maxima. Although I must say that I've had found SageMath as something a bit more if not approachable than at least easier to build my own modules with. It's free and open source, just as Maxima, but written in Python instead of Common Lisp making the barrier to entry for people who like to get their hands dirty in internal code a lot lower.

However CAS and similar software still has a lot to 'learn' when it comes to integration, especially in domain of non-standard ODE and PDE. Hardly a week goes by where I'm not encountering a case that my colleagues and I can solve on paper via human laziness and tricks that makes Mathematica grid beg for more resources and mercy ;). Here is almost a perfect example of what I'm talking about.

 

[bcrowell]

@DevacDave - I believe Sage's facilities for integration are just a front end for Maxima.

 

[DevacDave]

Possibly, it's possible that they use same shared libraries that originate form Maxima or just use it shamelessly through some API. To be honest it has been a pretty long time since I was involved with Sage. I can't deny that I might have confused something or operate under obsolete information. I do recall attempt on independent integration engine on Sage part, but it would not surprise me if people would abandon it for sake of not reinventing the wheel ;).

Either way, both are really good; I'm just favouring and 'advertising' Sage because I'm using it more often. In the end, excuse possible mistake on my part. It was not intended as some badly disguised attempt on discrediting Maxima, just sloppy fact-checking on my part.

 

[DaveC49]

Abramowitz and Stegun, Handbook of Mathematical Functions, used to be pretty good but as noted above Maxima /Mathematica/Sage probably have most of the numerical results from it built in.