The fundamental theorem of integral calculus

The fundamental theorem connects the two central operations of analysis, differentiation and integration, through the integral function.

Theorem — Fundamental theorem of integral calculus

If $f$ is continuous on $[a,b]$, the integral function $F(x) = \int_a^x f(t)\,dt$ is differentiable and its derivative is $F'(x) = f(x)$.

Consequence: $F$ is an antiderivative of $f$, and therefore $\boxed{\int_a^b f(x)\,dx = F(b) - F(a)},$ where $F$ is any antiderivative of $f$.

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Topics: Calculus theorems
Concepts: Integral function · Antiderivative · Fundamental theorem of calculus
Skills: Integrating