The other part of the Fundamental
Theorem of Calculus (FTC 1)
also relates differentiation and integration, in a slightly
different way.

Fundamental
Theorem of Calculus (Part 1)

If $f$ is a continuous function on $[a,b]$, then the
integral function $g$ defined by $$g(x)=\int_a^x
f(s)\, ds$$ is continuous on $[a,b]$, differentiable on
$(a,b)$, and $g'(x)=f(x)$.

What we will use most from FTC 1
is that $$\frac{d}{dx}\int_a^x
f(t)\,dt=f(x).$$

This says that the derivative of the integral
(function) gives the integrand; i.e. differentiation
and integration are inverse operations, they cancel
each other out. The integral function is an anti-derivative.

In this video, we look at several examples using FTC 1.

This video will show you why FTC 1
makes sense.

Notice: The notation $\int f(x) \,dx$, without any upper and lower
limits on the integral sign, is used to mean an anti-derivative of
$f(x)$, and is called the indefinite
integral. This means that
$\int\cos(x)\,dx=\sin(x)+c$, and we don't have to use the capital
$F$ any longer.