Proving only certain methods of Integration work on certain problems

Question

I have not studied Pure Maths (Stuff which includes rigorous proving) yet however I did take classes on Integration.
There were certain problems which we could do only by certain ways to get the results for instance the integral of $ln(x)$ can be easily computed via Integration by Parts however there is no meaningful substitution in my knowledge which can do the same.
I pondered a bit and then it seemed pretty obvious that not all integrals can be solved by all techniques however this is a very general statement to make and rigour is everything in mathematics so are there ways to prove such a statement?
I thought of a way where I tried using Proof by Contradiction and said if the statement is true then there should exist no counterexamples then took up the integral of some function where according to my knowledge there is no substitution technique like $ln(x)$ so the proof became contingent on the fact that there is no subsitution for the following integral which can solve it but couldn't think of any ways to do that

KCd · Answer

There certainly is a substitution that works: $u = xln x - x$. Then $du =ln x,dx$, so $int ln x,dx = int du = u = xln x - x$.
Of course I made that substitution because I knew the answer, but my point is that "not yet knowing the answer" isn't a mathematical concept. I gave you a substitution that worked, case closed.
Rather than try to show certain methods of integration don't work, a more robust concept is showing that the antiderivative of a (continuous) function does not lie among a certain class of functions, such as what are called the "elementary" functions.  See https://en.wikipedia.org/wiki/Liouville%27s_theorem_(differential_algebra). This is not going to tell you anything about how you can or can't work out an antiderivative of $ln x$ by techniques taught in a calculus course (in fact you know that it can be done), but it will tell you that functions such as $e^{-x^2}$ do not have an antiderivative among the kinds of functions students learn before taking calculus (the "elementary functions").

Proving only certain methods of Integration work on certain problems

One Answer

Add your own answers!

Ask a Question