Indefinite integrals with Wolfram Alpha - my test cases fail sometimes - why?

Question

I'd like to learn to use Wolfram Alpha to find indefinite integrals, but I'm not getting expected results for even simple test cases. I must be doing something wrong but I don't know what.

integrate exp(-a(y^2+x^2)) dx and integrate exp(-a(1+x^2)) dx fail:

but integrate exp(-ay^2)exp(-ax^2) dx and integrate exp(-a)exp(-ax^2) dx give me apparently correct answers:

Answer 1

This is because WA interprets a as a function in the failed examples, not a constant. Who knows why. (I found this by requesting the Mathematica input form from within Mathematica.)

Just put a * sign after the a to fix it.