# Thread: Integration in elementary terms

1. ## Integration in elementary terms

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?

2. Originally Posted by TheCoffeeMachine
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?
I don't think it matters because as you pointed out

$\int ( f(x) + g(x) ) dx = \int f(x)dx + \int g(x) dx$

In which case if $\int g(x) dx$ cannot be expressed in terms of elementary functions, neither can $\int (f(x) + g(x))dx$

I don't see how the addition of a function would change anything. Multiplication on the other hand...helps tons

3. Originally Posted by AllanCuz
I don't see how the addition of a function would change anything. Multiplication on the other hand...helps tons
That's what I thought Allan, but a completely unrelated reasoning made me suspect. Consider the number $\sqrt{2}$, an irrational number, and the number $2-\sqrt{2}$, another irrational number. If you add
them, $$\sqrt{2}+(2-\sqrt{2}) = 2$$, which is a rational number. So addition is nasty (as is her twin) and cannot be trusted. Now, take $f(x) = \frac{1}{\log{x}}[/tex] and $$g(x) = -\frac{1}{\log{x}}+x^2+3x+4$. Integral of $f(x)$ is non-elementary and the integral of $g(x)$ is non-elementary. If we add them, $f(x)+g(x) = x^2+3x+4$, the integral of which is of course elementary. This question came to my head while I was pondering over this: $\displaystyle \int\frac{n^2\sin^{2n-1}{nx}\cos{nx}}{\sqrt{\sin^{2n}{nx}+\cot^{n-1}{nx}}}\;{dx}$ No luck yet. 4. Originally Posted by TheCoffeeMachine That's what I thought Allan, but a completely unrelated reasoning made me suspect. Consider the number $\sqrt{2}$, an irrational number, and the number $2-\sqrt{2}$, another irrational number. If you add them, [tex]\sqrt{2}+(2-\sqrt{2}) = 2$$, which is a rational number. So addition is nasty (as is her twin) and cannot be trusted. Now, take $f(x) = \frac{1}{\log{x}}[/tex] and $$g(x) = -\frac{1}{\log{x}}+x^2+3x+4$. Integral of $f(x)$ is non-elementary and the integral of $g(x)$ is non-elementary. If we add them, $f(x)+g(x) = x^2+3x+4$, the integral of which is of course elementary. This question came to my head while I was pondering over this: $\displaystyle \int\frac{n^2\sin^{2n-1}{nx}\cos{nx}}{\sqrt{\sin^{2n}{nx}+\cot^{n-1}{nx}}}\;{dx}$ No luck yet. Fair point but I would only consider this possible when the addition of the two functions eliminates the source of the non-elementary problem DIRECTLY. I mean in your $g(x)$ you've included $-f(x)$. So in reality, your addition is $g(x) - f(x) + f(x)$ so if f(x) is the problem, of course it will cancel out and the integral of g(x) will be able to be expressed in elementary terms. And wow, where did you dig up that beast of an integral? I wouldn't have the first clue where to start...I'd message simplependulum and see if he's up for it (that guy is way to good at integrals!). 5. Originally Posted by TheCoffeeMachine That's what I thought Allan, but a completely unrelated reasoning made me suspect. Consider the number $\sqrt{2}$, an irrational number, and the number $2-\sqrt{2}$, another irrational number. If you add them, [tex]\sqrt{2}+(2-\sqrt{2}) = 2$$, which is a rational number. So addition is nasty (as is her twin) and cannot be trusted. Now, take $f(x) = \frac{1}{\log{x}}[/tex] and [tex]g(x) = -\frac{1}{\log{x}}+x^2+3x+4$. Integral of
$f(x)$ is non-elementary and the integral of $g(x)$ is non-elementary. If we add them, $f(x)+g(x) = x^2+3x+4$, the integral of which is of course elementary.

This question came to my head while I was pondering over this:

$\displaystyle \int\frac{n^2\sin^{2n-1}{nx}\cos{nx}}{\sqrt{\sin^{2n}{nx}+\cot^{n-1}{nx}}}\;{dx}$

No luck yet.
Mathematica 7 couldn't even do with non-elementary functions. You need to be very clever in how you choose to integrate in order to have it integrate to something known. Where did you come up with an integral like this?

6. Originally Posted by TheCoffeeMachine
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?
The answer is "yes" for exactly the reasons you state in your response to post #2.

Let $f(x)= x$ whose integral can be expressed in (very!) elementary terms. Let $g(x)= e^{x^2}$ whose integral cannot. Finally, let $h(x)= f(x)- g(x)= x- e^{x^2}$

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 $h(x)+ g(x)= f(x)- g(x)+ g(x)= x$ which has an elementary integral.