Let f(x) and g(x) be two functions whose antiderivatives exist, such that the integral of f(x) can be
expressed in elementary terms while the integral of g(x) can cannot be. Is it possible that the integral
of f(x)+g(x) can be expressed in elementary terms? Perhaps not because of the linearity of integration?
That's what I thought Allan, but a completely unrelated reasoning made me suspect. Consider the number , an irrational number, and the number , another irrational number. If you add
them, [tex]\sqrt{2}+(2-\sqrt{2}) = 2[/tex], which is a rational number. So addition is nasty (as is her twin) and cannot be trusted. Now, take . Integral of
is non-elementary and the integral of is non-elementary. If we add them, , the integral of which is of course elementary.
This question came to my head while I was pondering over this:
No luck yet.
Fair point but I would only consider this possible when the addition of the two functions eliminates the source of the non-elementary problem DIRECTLY.
I mean in your you've included . So in reality, your addition is
so if f(x) is the problem, of course it will cancel out and the integral of g(x) will be able to be expressed in elementary terms.
And wow, where did you dig up that beast of an integral? I wouldn't have the first clue where to start...I'd message simplependulum and see if he's up for it (that guy is way to good at integrals!).
The answer is "yes" for exactly the reasons you state in your response to post #2.
Let whose integral can be expressed in (very!) elementary terms. Let whose integral cannot. Finally, let
Now, it is true that neither of g nor h have integrals that can be expressed in terms of elementary functions but their sum is which has an elementary integral.