Well folks, here it is: the fundamental theorem, that ties
together integral and differential calculus. It comes in two parts:
-
If $f$ is continuous on $[a,b]$, then
$\int_{a}^{b}f(x)dx=F(b)-F(a)$
where $F$ is any antiderivative of $f$; that is, a function such that $F'=f$.
- If $f$ is continuous on $[a,b]$,
then the function $g$ defined by
$g(x)=\int_{a}^{x}f(t)dt \;\;\;\;\; a \le x \le b$
is continuous on $[a,b]$ and differentiable on $(a,b)$, and $g'(x)=f(x)$.
It says that to integrate, all you have to do is think of an antiderivative
F of the integrand f. Note that $g(a)=0$ -- so $g$ is that
particular anti-derivative $F(x)$ of $f(x)$ which is zero at $a$.
Note the only essential condition:
- $f$ is continuous on $[a,b]$.
Have a look at Example 8, p. 316.