kolleamm said:
Online it says that the integral is the opposite of the derivative. So x^2 is the integral of 2x.
So if f(x) = x^2 , does that mean that the integral is just the function itself? Basically whatever f(x) equals?
Thanks in advance
You're question is a bit confusing. So let's start with a smooth function ##f: \mathbb{R} \rightarrow \mathbb{R}##, e.g. ##f(x)=x^2##.
(I picked smoothness here for reason of simplicity and it means mathematically basically the same as in ordinary language: no gaps or corners, pretty smooth.)
Then the derivative ##f\,' ## of ##f## is again a smooth function that associates to each point ##x## the slope of ##f## which is defined as the steepness of its tangent, i.e. the steepness of the touching line. In the example we have ##f\,'(x)=2x## which means, that e.g. at the point ##x=5## we have a slope of ##2\cdot 5 = 10## (##10## inches of gained height on ##1## inch horizontal deviation).
The integral ##\int f(x)\, dx = F(x)## of ##f## is a function ##F## that measures the area between the ##x-##axis and the function ##f##. Basically as we measure the area of a room: length times width. Only that it also works for curved walls. In the example we have ##\int x^2 dx = \frac{1}{3}x^3## which means, that the area ##A## underneath the parabola ##x^2##, say between the points ##x=1## and ##x=4##, is ##A= \frac{1}{3}\cdot 4^3 - \frac{1}{3} 1^3 = 21##.
Now what makes a derivative and an integral sort of "opposite" is, that ##\int f\,'(x) dx = f(x)##, or in our example ##\int 2x \,dx= x^2##.
