Theorem: Fundamental Theorem of Calculus 

Let $I$ be a real interval and let $F:I\to\mathbb R$ be an antiderivative of a continuous function $f:I\to\mathbb R$. Then, for all $a,b\in I$ the following holds for the Riemann integrals of $f$ on the closed real interval $[a,b]$:

$$\int_a^bf(x)dx=F(b)-F(a).$$

Different Notation

$$\int_a^bf(x)dx=F|_a^b.$$

| | | | | created: 2017-08-03 13:40:17 | modified: 2017-08-03 13:46:57 | by: bookofproofs | references: [581]

1.Proof: (related to "Fundamental Theorem of Calculus")