I have a couple of questions regarding hyperbolas and their integrals. If it's too much, don't feel like you have to answer all 3 questions.
My first question:
The integral of a function like 1/x^2 is -1/x - I sufficiently understand the algebraic manipulations that gets us to this result, but geometrically (and in terms of intuitive understanding) I am at 0. How can the integral of such a function (which is always above 0) be a negative number?
Is it due to the area between -1 and 1, so that you go "beyond" infinity and you end up with a negative - sort of like the idea that 1+2+3+4+… = -1/12 (the true logic behind which I of course can't appreciate).
The clue that makes me think all this is that as you add more and more positive area (as x increases for the indefinite integral of 1/x^2), the positive additions reduce the "infinite" negativity (that exists, as theorized, due to the first bit of the function around x=0) , hence why -1/x approaches 0. To get to 0, you would have to take the indefinite integral at x=infinity, which can be geometrically "explained". All this mumbo jumbo is also somewhat consistent with the fact that the definite integral of such a function (1/x^n) is a positive number as F(b) - F(a) is a positive (thank god)!
My second question:
What is the definite integral of 1/x^2 between x=1 and x=infinity related to (pi^2)/6 (the basel problem.
Finally (and most importantly):
Does calculus eventually become second nature and perfectly intuitive like arithmetic?