The Fundamental Theorem of Calculus (Part 1)

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.