Fundamental Theorem of Calculus

Definition

differentiating a function and integrating a function are inverses of each other apart from a constant value.

First Fundamental Theorem of Calculus

Suppose $f\in \mathcal{R}[a,b]$, and $F$ is the Indefinite Integral of $f$ on $[a,b]$ $F(x)={\int }_a^{x}f(t)dt\forall x\in [a,b]$

• $F$ is uniformly continuous on $[a,b]$
• If $f$ is continuous on $[a,b]$, then
  • $F$ is differentiable on the $[a,b]$ and $F^{\prime }(x)=f(x)\forall x\in [a,b]$

Second Fundamental Theorem of Calculus

Suppose $f\in \mathcal{R}[a,b]$, and $F:[a,b]\to \mathbb{R}$ is continuous on $[a,b]$ and differentiable on $(a,b)$. $F^{\prime }(x)=f(x)\forall x\in [a,b]$ If $F$ is Antiderivative of $f$, then ${\int }_a^{b}f(x)dx=F(b)-F(a)$