I am a high school student and am a beginner in integral calculus. In one of my reference textbooks is said that there were certain integrals which “can't be found”.
Some of these include
${\int}{\sin x\over x}\ {\rm dx}$, ${\int}{\cos x\over x}\ {\rm dx}$, ${\int}{1\over \log x}\ {\rm dx}$, ${\int}x\tan x \ {\rm dx}$
I graphed these equations in Desmos and found nothing strange. None of the explanations online made any senese and I failed to understand the following-
1)What about these functions makes them non-integrable?
2)Are there infinitely many functions like this?
3)Why does this happen when the curve is continuous and the area is well defined?
Best Answer
Those integrals can be found. You can compute them with any precision that want. The problem is that you cannot express them using only elementary functions. This is a very broad class of functions which include probably any differentiable function that you've ever heard of.