Fundamental Theorem of Calculus, Part 1

Differentiating an accumulation function $g(x)=\int_a^x f(t)\,dt$ just returns $f(x)$ .

The Fundamental Theorem of Calculus, Part 1 states that if $g(x)=\int_a^x f(t)\,dt$ , then $g'(x)=f(x)$ wherever $f$ is continuous. In words: differentiation and integration are INVERSE operations — accumulating $f$ 's values via an integral and then differentiating the result just hands back $f$ itself, with no work beyond substitution.

Examples

For $g(x)=\int_2^x t^3\,dt$ , FTC Part 1 gives $g'(x)=x^3$ immediately, with no need to evaluate the integral first.