Fundamental Theorem of Calculus

• First Fundamental Theorem of Calculus

  If $f$ is continuous on the interval $[a,b]$ and $F$ is an antiderivative of $f$ on the interval $[a,b],$ then

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

• Second Fundamental Theorem of Calculus

  If $f$ is continuous on an open interval $I$ containing $a,$ then for every $x$ in $I,$

  $$\frac{d}{dx}\left[\int_a^x f(t) dt\right]=f(x)$$