General formula for integration

>>8467226
There and it is the formula for the integral of ax^n

What you have to understand is that transcendental functions (ones that cannot be expressed as polynomials) are really random and almost don't even make sense.

In Set Theory we consider functions to be any set of ordered pairs of the form (a.b) such that if (a.b) is part of the function then (a.c) for any c not equal to b must NOT be in the set.

Like this we make sense of transcendental functions that really make no sense from a traditional algebraic perspective. To understand this I ask you, what generates the values of the sine function? And I don't mean the approximations our calculators, I mean what actually generates them? A bunch of fucking triangles.

Does it make sense to take the integral of a triangle? Well, not really. But our theory of limits (which came way after the integral) allows us to make sense of stuff like the integral of cosx is sinx.

Now that we understand that transcendental functions are really weird and actually arbitrary (could be literally anything) then it makes sense to approximate our well behaved transcendetals as infinite polynomials and work with them like that. That is how computers do it and that is how you will have to do it from time to time because if you don't then nothing makes sense about these functions.

So there is a general formula for integrals and that is the integral of a polynomial, as those are the fundamental (and arguably the only) algebraic objects.

>>8467226
>So there is a general formula for integrals and that is the integral of a polynomial, as those are the fundamental (and arguably the only) algebraic objects.
Why bother with other methods then?

>>8467226
Take a look at differential Galois theory. It deals with the question of which indefinite integrals can be expressed explicitly in terms of known functions.

>>8467276
>transcendental functions (ones that cannot be expressed as polynomials) are really random and almost don't even make sense.
You're a fucking retard. Or a master baiter.

>>8467276
a faulty argument with a correct conclusion.

classic algebraist trying to explain analysis

>>8467226
>Will there ever be a general formula for solving integrals?
riemann sums

>>8467352
this

and numerical methods

flatten curve, do length * width

>>8467352
you mean lebesgue sums

Why not expressing a function in terms of any other functions?
Are there good and bad choices?

[math]f(x)=\sum g(n)(x)[/math]

with whatsoever [math]g[/math].

>>8468106
you want your g's to be orthogonal with respect to some sort of operator, so you can extract meaningful information out of them.

You also want the sequence of partial sums to be cauchy so you can use them as good approximations.

>>8468116
Thank you.

>>8467226
F(b) - F(a)

>>8468129
/thread