# Difference between first and second fundamental theorem of calculus

In first fundamental theorem of calculus,it states if $$A(x)=\int_{a}^{x}f(t)dt$$ then $$A'(x)=f(x)$$.But in second they say $$\int_{a}^{b}f(t)dt=F(b)-F(a)$$,But if we put $$x=b$$ in the first one we get $$A(b)$$.Then what is the difference between these two and how do we prove $$A(b)=F(b)-F(a)$$?

• If $f$ is continuous, then the first theorem's consequent is true, which in turn guarantees that the second theorem's consequent is also true. If on the other hand $f$ is not continuous, then the first theorem is inapplicable; if $f$ is integrable and has a primitive (despite not being continuous), then the second theorem's consequent is true. p.s. There exist functions that are integrable but have no primitive. – Ryan G Feb 14 at 15:16
• How do we prove $A(b)=F(b)-F(a)$ under valid assumptions – Aritra Barua Feb 14 at 15:23
• Check out these, previous discussions, and my favourite. – Ryan G Feb 14 at 15:53

In the first part you mentioned, $$f$$ is assumed to be continuous. In the second part, $$f$$ can be assumed only Riemann integrable on the closed interval $$[a,b]$$. When $$f$$ is continuous, the second part indeed follows from the first part.
• @AritraBarua: if you use $A(b)=F(b)-F(a)$, then you are using the first part, where $f$ is assumed to be continuous. – user9464 Feb 14 at 16:28