Calculus

**First fundamental theorem of Calculus:**

If $f$ is continuous on the closed interval $[a, b]$ and $F$ is an antiderivative of $f$ on $[a,b]$, then the definite integral from $a$ to $b$ of $f(x)dx$ is equal to the difference of $F(b)$ and $F(a)$.

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

**Second fundamental theorem of Calculus:**

If $f$ is continuous on an open interval $I$ containing $a$, then for every $x$ in the interval, the derivative with respect to $x$ of the definite integral from $a$ to $x$ of $f(t)dt$ is equal to $f(x)$.

$\frac{d}{dx} [\int_a^x \! f(t) \, dt] = F(x)$