Fundamental Theorems of Calculus

The first fundamental theorem of calculus states that, if $f$ is Continuous on the Closed Interval $[a,b]$ and $F$ is the Antiderivative (Indefinite Integral) of $f$ on $[a,b]$, then

$$\int^b_a f(x)\,dx = F(b)-F(a).$$ (1)

The second fundamental theorem of calculus lets $f$ be Continuous on an Open Interval $I$ and lets $a$ be any point in $I$. If $F$ is defined by

$$F(x) = \int^x_a f(t)\,dt,$$ (2)

then

$$F'(x) = f(x)$$ (3)

at each point in $I$.

The complex fundamental theorem of calculus states that if $f(z)$ has a Continuous Antiderivative $F(z)$ in a region $R$ containing a parameterized curve $\gamma: z = z(t)$ for $\alpha \leq t \leq \beta$, then

$$\int_\gamma f(z)\,dz = F(z(\beta))-F(z(\alpha)).$$ (4)

See also Calculus, Definite Integral, Indefinite Integral, Integral