Q:

How to use integrals to evaluate a function?

Textbooks often show:
$f(x)=\int f'(x)\ dx$
But how do we evaluate $f(x)$ at a specific x, let's say $f(2)$?
Let's say
$f(x)=x^2 ,\     f'(x) = 2x$
then $f(3) = 9$
but on the right hand side, shall we do $\int_{-\infty}^3\ f'(x)\ dx$? That doesn't work...
Thanks in advance.

A:

When you find a definite integral, you have to have upper and lower limits. Visually, your $f'(x)$ is a graph and the integral of that is the area under the graph. So to actually calculate the area, you (usually) need to cut it off on the left and the right: this is a and b in the picture below.

Also remember, if you take the derivative of f(x) (and the derivative exists) you will get one valid function, but integrating f'(x) gets you an infinite number of functions that all differ by a constant. This is the indefinite integral; for example, $\int2xdx$ can get you $x^2$, $x^2 +1$, $x^2 + 2$, and so forth.

