Fundamental Theorem of Calculus problem

Since the function $\,t^3+1\,$ is continuous (and derivable) everywhere, it has a primitive function $\,G(t)\,$in any finite interval. Using the FTC , write

$$\int_a^x(t^3+1)dt=G(x)-G(a)\\\Longrightarrow \frac{d}{dx}{\left(\int_a^x(t^3+1)dt\right)}=\frac{d}{dx}{(G(x)-G(a))}=G'(x)=x^3+1$$

The Fundamental Theorem of Calculus doesn't talk about geometrical results, but about the "fundamental" relation between the operation of integration and that of differentiation. Namely, it says the following:

THEOREM. Let $f$ be a continuous over $[a,b]$. Define $F$ on $[a,b]$ by

$$F(x)=\int_a^x f(t) dt$$

Then $F$ is differentiable, and $F'(x)=f(x)$.

The corollary is

COROLLARY Let $f$ be continuous over $[a,b]$ and $f=g'$ for some $g$.

Then

$$\int_a^b f(t)dt=g(b)-g(a)$$

Note we can find this reversed in the books (One is the theorem and the other the corollary, or one is called FTC 1 and the other FTC 2). I recommend you read these two questions some users already asked about FTC:

• Understanding the relation between differentiation and integration.

• How do explain FTC to my teacher?