# variable bounds on definite integrals in Wolfram|Alpha

I often have trouble getting Wolfram|Alpha to compute definite integrals with variable bounds, and I'm wondering whether I'm missing something.

Some examples: $\int_a^b \sin x\mathrm dx$ and $\int \arcsin x \mathrm dx$ work fine; $\int_0^a \arcsin x \mathrm dx$ already requires extra time to finish; and $\int_a^b \arcsin x \mathrm dx$ doesn't compute even with extra time.

I'm thinking it probably has something to do with the conditions on the bounds, since there are conditions for the arcsine but not for the sine. It could be that W|A is having a hard time figuring out the conditions by itself; if so, how could I help it along?

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I usually write something like "integrate xdx with 0<x<2" – Asaf Karagila Jul 13 '11 at 12:28
But that specifies the bounds; I want to specify conditions/assumptions on the bounds. – joriki Jul 13 '11 at 12:35
I did not try your case but I had (and stil have) W|A behaving differently in Firefox and Internet Explorer. – Alessandro Jacopson Sep 23 '11 at 6:39

Yes, in Mathematica directly specifying conditions reduces computation time and sometimes allows to obtain a result at all. Mathematica input Integrate[ArcSin[x],{x,a,b},Assumptions->{-1<=a<b<=1}] gets a result, but somewhat strange one, the integral is considered as indefinite.

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Thanks -- it seems W|A can't handle that input -- do you happen to know how I can specify such assumptions to W|A? – joriki Jul 13 '11 at 12:37
That's very strange. I did press the link. I tried it again now, and I got the correct result, and then a few seconds later it gets replaced with a large list of suggestions what to try ("different phrasing or notations" etc.) Does the result stay around in your browser? – joriki Jul 13 '11 at 12:43
Bizarre -- I tried it again several times, and around the third time the result stayed. – joriki Jul 13 '11 at 12:46
@joriki: Well, very strange indeed. I get the correct result and there is a progress bar and after some progress it starts complaining. Also there is a mysterious "+constant" term appearing in the otherwise correct solution -- well, you can add some constants and get a correct result, but definitely not arbitrary ones :) [ Added I couldn't get the result to stay so far] – t.b. Jul 13 '11 at 12:49
@joriki I have Opera, result stays. The correct form seems to be Assuming[-1<=a<b<=1,Integrate[Sin[x],{x,a,b}]] because it produces an Input field in the result which can be copied as Mathematica plaintext. But putting ArcSin[x] instead of Sin[x] here does not gets anything, probably due to time limit. – Andrew Jul 13 '11 at 13:22