I have trouble understanding formally stated mathematical definitions and theorems. However, I am still trying to memorize both parts of the Fundamental Theorem of Calculus. Could someone please provide a short example of the first part?
I understand textbooks will often switch part 1 and part 2, but for the sake of this question, we'll say part 1 is this.
If $f(x)$ is continuous on $[a,b]$, then the function $g(x)$ defined by
$$g(x)=\int^x_af(t)dt \;\;\;\;\;\;\;\;\;\;\; a \le x\le b$$
is continuous on $[a,b]$ and differentiable on $(a,b)$ and $g'(x)=f(x)$.
The fundamental theorem of the calculus relates integration to differentiation.
The first part essentially says that the derivative of an anti-derivative is the original function.
For example, if $f(x)=x,$ then we can define $g(\color{blue}x)=\int_a^\color{blue}x t\;dt=\frac12(x^2-a^2),$ and $g'(x)=x.$
Make sure to write $x$ as a bound of integration, not part of the integrand.