Why can't some integral be"found" though they are anti-derivative & exist?

Question

In my book, a list of integrals have been given which the author states

... such anti-derivatives "cannot be found".

Some of the members of the list are as under:

$\int\dfrac{\sin x}{x} dx$ , $\int\dfrac{1}{\log x} dx$, $\int\sqrt{1 - k^2\sin^2x}dx$, $\int\sqrt{\sin x}dx$, $\int\cos(x^2) dx$, $\int x\tan xdx$ , $\int e^{-x^2}dx$, $\int\dfrac{x^2}{1 +x ^5}dx$, $\int\sqrt{1 + x^3}dx$.

The author only mentions that:

...not every anti-derivatives, even when it exists, is expressed in closed form ....

Now, can anyone tell me why these integrals "cannot be found"? Are they discontinous, non-differentiable or what? Also, what did the author mean by closed form??

Answer 1

Remember that a function is simply a rule for turning an input into an output. This means that: $$f(x)=\int_0^xe^{-t^2}dt$$ is a perfectly fine function. The fundamental theorem of calculus tells us that $f(x)$ is an antiderivative of $e^{-x^2}$.

However, $f(x)$ cannot be written in closed-form. What that means, in this context, is that $f(x)$ cannot be written as some combination of polynomials, trig functions, exponents, logarithms, additions, subtractions, multiplications, and divisions. (A function that can be written in such a form is usually called an elementary function. $e^{-x^2}$ does not have an elementary antiderivative.)

The proof that $e^{-x^2}$ has no elementary antiderivative is extremely difficult.

Answer 2

It means that these antiderivatives exist and are perfectly fine functions, but they can not be expressed by other elementary functions (functions that we are already well familiar with). If such anti derivative comes up very often, it is often given a name. Then you can study its properties and use it like any other function. Such functions are often called "special functions", but there is nothing more special about them than about sine or exponential.

Answer 3

Closed form means that the integral cannot be expressed in standard mathematical functions like trigonometric, exponential or polynomial functions etc.
See the example of $\int {{2^2}^2}^x dx$. Also, you may want to study elliptic integrals.

Answer 4

Can anyone tell me why these integrals “cannot be found” ?

For instance, the roots of a $($general$)$ quintic polynomial cannot be found either, i.e., they cannot be expressed in closed form. Basically, what the Abel-Ruffini theorem is for polynomials, Liouville's theorem and the Risch algorithm are for calculus.