Solving Difficult Integrals: Any Ideas for This Tricky Integral?

[Schrodinger's Dog]

Ok I'm sure everyone is bored to death by this by now, but I'm finding it quite interesting and quite informative for someone who's forget a lot about integration, so with that in mind it might be of use to someone, here's how he justifies the a in the integral and calling it "definite".

(2)... The point about the constants is an important one. If, for example, you have I = e^{ax}dx, then you can effectively get rid of the constant, a, from being inside the integral, by simply substituting for, say, t = ax. Differentiating this, we get dt = a dx. Rearranging, therefore dx = dt/a. Substitute in the integral, for ax, and dx, and we now get,
I = (e^{t}dt)/a .We divide I by the factor a, but a has now come outside the integral of (e^{t}dt) .

(3)... We can do a similar substitution for pretty well any integral.So in general, if I = f(ax)dx, where f is some function, then I=(f(t)dt)/a .

I can't see a problem with this, but then I'm seriously not that good, or that au fait atm with it all, although it's coming back gradually.