How do I use the fundamental theorems of calculus to prove the following statements?
$\frac{d}{dx} \int f(x)dx = f(x)$
$\int \left( \frac{d}{dx} f(x) \right) dx = f(x) + C$
I've tried to combine the first and second theorems together, however, I can only end up with definite integrals.
I've been trying to prove this as it was mentioned in James Stewart's Calculus textbook (8th edition):
Taken together, the two parts of the Fundamental Theorem of Calculus
say that differentiation and integration are inverse processes. Each undoes what the
other does.
However, there is no specific proof about this. Can someone help me with this proof?
Best Answer
For starters, the statement of the fundamental theorem of calculus you present is not accurate. The fundamental theorem of calculus is specifically a statement about Riemann integrals, or extensions thereof, and it only applies when the function being integrated satisfies certain conditions.
First part of the theorem: Let $f$ be a function on $[a,b]$ that is continuous almost everywhere. Then $$\frac{d}{dx}\int_a^xf(t)dt=f(x)$$ for every $x$ on $]a,b[$.
Second part of the theorem: Let $f$ be a function on $[a,b]$ that is Riemann integrable. If there exists some $F$ on $]a,b[$ such that $\frac{dF}{dx}=f$, then $$\int_a^bf(t)dt=F(b)-F(a).$$
The first part of the theorem states, rather informally, that the derivative "inverts" integration. The second part of the theorem states, rather informally, that the integral "inverts" differentiation.
The second part of the theorem can be more suitably expressed as, let $f$ be a function on $[a,b]$ that is differentiable, and let $f'=\frac{df}{dx}$ be Riemann integrable on $[a,b]$. Then $$\int_a^b\frac{df}{dx}(x)dx=f(b)-f(a).$$
Anyway, as you can see, neither of the statements you presented in your post is actually part of the fundamental theorem of calculus. Your first statement is merely saying: if there exists some $F$ on $]a,b[$ such that $\frac{dF}{dx}=f$, then $F$ is such that $\frac{dF}{dx}=f$ on $]a,b[$, which is obviously true. Your second statement is saying that, if $f$ is differentiable on $]a,b[$, then $\frac{d}{dx}(f)=\frac{d}{dx}(f + C)$, which is also obviously true, albeit less so; and neither of these has any particular relationship to the fundamental theorem of calculus, as they are just obvious statements.