The method of judging whether indefinite integral can be made into elementary function involves "deep mathematical knowledge". Generally speaking, you only need to remember a few common non-integral functions, e x? ，cosx/x，sinx/x，sin(cosx)，√sinx。 ，sin(x？ ) and so on. There is also a small rule that they are all composed of different kinds of elementary functions.

In fact, integrable functions only account for a small part of elementary functions.

If you really want to judge for yourself, you can use the following methods to prove part of it:

T = sinx is transformed into binomial differential form, and Chebyshev theorem is used to judge whether it can be expressed as an elementary function.

Chebyshev theorem: the binomial differential equation ∫ x p (1+x r) q dx (where a and b are not equal to 0, p, q and r are rational numbers) can be expressed as an elementary function if and only if it is q, (p+ 1)/r, (p

In addition, there are joseph liouville's third and fourth theorems, differential Galois theory and other methods to prove that the original function is not an elementary function:

Literature can be found in Twenty Lectures on Mathematical Analysis, journal of hulunbeier college No.2, 2005, Indefinite Integral and its Proof (English version) of Baidu Library, Galois' theory of linear differential equations, etc.