\int_a^b f(x)\,dx=F(b)-F(a),\quad F'=fVariables
- f: integrand
- F: any antiderivative of f
- a, b: integration limits
How to use this formula
Connects definite integration with antiderivatives.
Important notes
- The function must satisfy the theorem’s continuity conditions on the interval.
- Evaluate the antiderivative at the upper limit minus the lower limit.
Quick example
∫₀¹ 2x dx = [x²]₀¹ = 1.
Applicability, worked calculation, and verification
Domain and applicability
Apply Fundamental Theorem of Calculus only when the listed variables are defined, denominators are nonzero, and the problem satisfies the assumptions stated on this page.
Assumptions and domain checks
- The function must satisfy the theorem’s continuity conditions on the interval.
- Integration limits, the differential, and any constant of integration must be included where required.
- The function must satisfy the differentiability, continuity, or integrability conditions required by the operation being used.
Do not use this formula when
- Do not use Fundamental Theorem of Calculus without checking differentiability, integrability, convergence, interval, and transform-convention requirements.
Boundary and special cases
- For Fundamental Theorem of Calculus, check zero, negative, and extreme input values before relying on the result.
- When using Fundamental Theorem of Calculus, confirm that denominators, radicals, logarithms, and domain restrictions remain valid for the chosen values.
Equivalent and alternative forms
- Keep the canonical LaTeX form \int_a^b f(x)\,dx=F(b)-F(a),\quad F'=f for copying; rearrange only after preserving equivalence and domain restrictions.
Worked example
Evaluate ∫₀¹2x dx. An antiderivative is F(x)=x², so F(1)-F(0)=1.
- Choose F(x)=x² because F′(x)=2x.
- Evaluate the upper endpoint: F(1)=1.
- Evaluate the lower endpoint and subtract: 1-0=1.
Independent verification
A geometric check gives the area of a triangle with base 1 and height 2, which is also 1.
Common mistakes
- Keep the integration bounds and differential attached to Fundamental Theorem of Calculus, and include a constant of integration for an indefinite integral.
- Do not apply a differentiation or integration rule outside its domain or omit endpoint, continuity, or constant-of-integration checks.
Continue the workflow
Use Fundamental Theorem of Calculus in your own work
- Check the domainMatch the variables and assumptions to the problem before substituting values.
- Copy the exact notationPreserve grouping, signs, and exponents in
\int_a^b f(x)\,dx=F(b)-F(a),\quad F'=f. - Edit or convertOpen the expression in the LaTeX editor, then export it for your document or web page.
Review and verification
Last reviewed: 2026-07-26
Review method: Reviewed the notation, variable definitions, applicability conditions, worked workflow, and verification method for Fundamental Theorem of Calculus.
Verified against: OpenStax Calculus Volume 1
Automated quality check: Passed the core formula indexing gate.
Formula references
- Calculus Volume 1OpenStax, Rice University — Reviewed limits, derivatives, integrals, and foundational calculus formulas.
Frequently asked questions
What is the Fundamental Theorem of Calculus used for?
Connects definite integration with antiderivatives.
Can I copy this formula as LaTeX?
Yes. Copy \int_a^b f(x)\,dx=F(b)-F(a),\quad F'=f or open it in the LaTeX editor.
What should I check before using it?
Confirm that each variable, unit, domain restriction, and assumption matches the problem.