# Why is the integral the antiderivative of a function?

What's the demonstration that the antiderivative of a function is the integral?

• It's known as the fundamental theorem of calculus and you can look it up in rigorous treatments of calculus. – Michael Greinecker Nov 11 '12 at 23:11
• Any calculus text will have a proof. – Stefan Smith Nov 12 '12 at 16:11
• I upvoted this question because in my opinion, it's a real question because some mathematicians have a demand for rigour ad just like the Jordan curve theorem, it seems so obviously true that an integral is an antiderivative but is probably actually quite difficult to give a rigorous proof of. I actually was specifically looking for a question like this so I did a Google search to find it. – Timothy May 27 at 3:29

Here's the intuition. Suppose $f$ is continuous, and let $$F(x) = \int_a^x f(t) \, dt.$$ Let $\Delta x > 0$ be tiny. Then \begin{align*} F(x + \Delta x) - F(x) &= \int_x^{x+\Delta x} f(t) \,dt. \end{align*} But since $f$ is continuous, $f$ is approximately constant over the tiny interval $[x,x + \Delta x]$. Thus \begin{align*} \int_x^{x+\Delta x} f(t) \,dt &\approx \int_x^{x + \Delta x} f(x) \, dt \\ &= f(x) \int_x^{x + \Delta x} 1 \, dt \\ &= f(x) \Delta x. \end{align*}
As $\Delta x \to 0$, the approximation gets better and better, so we conclude that $$F'(x) = \lim_{\Delta x \to 0} \frac{F(x + \Delta x) - F(x)}{\Delta x} = f(x).$$