0
$\begingroup$

I'm going to take an example to elaborate my question.

My teacher said that - $\int \sin(x^3)dx$ is unsolvable for all x. Just recently, he said that $\int_{-\pi/6}^{\pi/6} \sin(x^3) = 0$. Now I get why its zero but my question is -

  • Indefinite Integration is an anti-derivative process. If a function can be differentiated, then why can't it be integrated so as to get its primitive? Are those functions like some "unsolved mysteries of mathematics" ? Because if we're integrating something then its almost understood that its primitive exists (at least thats what I think).
  • What about an expression makes it non-integrable?
  • Can we look (or examine it a little bit) and say that the expression is integrable or not? If so, what are the things we should look out for?
$\endgroup$
7
  • 3
    $\begingroup$ Definite integrals are not an anti-derivative process. Anti-derivatives can be one way to solve them, but there are lots of ways to solve them. Also, even when you can’t “solve” and indefinite integral, there is a function which is the indefinite integral. You just can’t write that function down in terms of known functions. $\endgroup$ Commented Aug 20, 2021 at 17:36
  • 8
    $\begingroup$ Whether an indefinite integral has a closed form is a highly technical area. That doesn’t mean the indefinite integral doesn’t exist, it just means it doesn’t have a certain form. Just as some real numbers are irrational, some indefinite integrals exist and can’t be written in a closed form in terms of other “usual” functions. In any even, these functions are integrable. The functions which are actually not integrable are highly strange, and in particular, are not continuous. $\endgroup$ Commented Aug 20, 2021 at 17:39
  • 3
    $\begingroup$ @ThomasAndrews 's comment is spot on. Let me complete it by saying that computing an antiderivative is akin to finding the value of a function on its entire domain. This is far, far more complex than guessing the value of said function at certain points only (which is what the definite integral on a given interval is). $\endgroup$ Commented Aug 20, 2021 at 17:39
  • $\begingroup$ Just to add, not every continuous function can be differentiated. There are continuous functions that are nowhere differentiable. $\endgroup$ Commented Aug 20, 2021 at 17:50
  • 1
    $\begingroup$ Point taken. My explanation was not so much about antiderivative vs derivative, but rather about antiderivative at all points vs at some points. $\endgroup$ Commented Aug 20, 2021 at 22:23

1 Answer 1

Reset to default
5
$\begingroup$

Whether an indefinite integral has a closed form is a highly technical area.

That doesn’t mean the indefinite integral doesn’t exist, it just means it doesn’t have a certain form.

Just as some real numbers are irrational, some indefinite integrals exist and can’t be written in a closed form in terms of other “usual” functions. In any event, these functions are integrable.

A useful non-integral example of a similar thing is the Lambert W-function. Given an $x\geq 0,$ $y=W(x)$ if $x=ye^y.$

We can’t write $W$ in terms of “elementary functions.” (Proving that requires some extremely tricky techniques.) But we can compute $W$ numerically, and we can tell specific values, like $W(0)=0, W(e)=1.$ We can prove things about $W.,$ like that it is an increasing function.

Mathematicians do this a lot. We can prove properties of $F(x)=\int\sin(x^3)\,dx,$ even if we can’t write $F(x)$ in a useful way.

There are functions which are actually not integrable, but they are highly strange, and in particular, those functions cannot be written in a closed form, either.

$\endgroup$
5
  • $\begingroup$ What do you mean by "closed form" ? $\endgroup$ Commented Aug 20, 2021 at 18:01
  • 1
    $\begingroup$ It means in terms of a composition of some set of known functions. The indefinite integral for $\sin(x^3)$ can be written as: $$\int_{0}^{x}\sin(t^3)\,dt +C,$$ but that isn’t a close form, because integrals are complex. The usual allowed “elementary” functions are addition, multiplication, division, constants, $e^x,$ logarithms, trig functions, addition, multiplication, polynomials, and inverses of these functions, where applicable. So $\tan^{-1}(e^x\ln x)+x^2$ is a closed form. $\endgroup$ Commented Aug 20, 2021 at 18:17
  • $\begingroup$ Okay. Thank you $\endgroup$ Commented Aug 20, 2021 at 18:33
  • 1
    $\begingroup$ A fairly simple example: The derivative of a polynomial is a polynomial so the derivative of a rational function is a rational function. But $\int (1/x)dx (x>0)$ is not a rational function. $\endgroup$ Commented Aug 20, 2021 at 18:46
  • $\begingroup$ Even though you brought it up, do not forget elementary functions vs non elementary functions. If you let one use any academically recognized functions, then you can find $\int \sin\left(x^3\right) dx$ in terms of the Generalized Exponential Integral function etc. $\endgroup$ Commented Aug 25, 2021 at 21:06

You must log in to answer this question.

Start asking to get answers

Find the answer to your question by asking.

Ask question

Explore related questions

See similar questions with these tags.