# Difference between first and second fundamental theorem of calculus In first fundamental theorem of calculus,it states if A(x)=int^x_a f(t) dt then A′(x)=f(x).But in second they say int^b_a 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)?

Difference between first and second fundamental theorem of calculus
In first fundamental theorem of calculus,it states if $A\left(x\right)={\int }_{a}^{x}f\left(t\right)dt$ then ${A}^{\prime }\left(x\right)=f\left(x\right)$.But in second they say ${\int }_{a}^{b}f\left(t\right)dt=F\left(b\right)-F\left(a\right)$,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)?
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Bestvinajw
They have different assumptions.
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.