CalcRef

Integrals · Fundamental Theorems

FTC Part 2

∫abf(x) dx=F(b)−F(a) where F′(x)=f(x)\int_a^b f(x)\, dx = F(b) - F(a) \text{ where } F'(x) = f(x)

The Fundamental Theorem of Calculus Part 2: A definite integral can be evaluated using any antiderivative F of f.

Conditions. f must be continuous on [a, b]. F is any antiderivative of f.
  • 0 variables
  • 2 worked examples
  • 19 in Integrals

Worked examples

Evaluate ∫₁³ 2x dx.
  1. Antiderivative: F(x) = x²
  2. F(3) - F(1) = 9 - 1 = 8

Answer: 8

1 more worked examplePremium

Evaluate ∫₀^π sin x dx.

Step-by-step solution locked.

Unlock the full working. CalcRef Premium opens the step-by-step solution above, and 141 worked solutions across the whole library - the intermediate lines, not just the answer. The formula statement, its conditions, its variables and the first worked example stay free on every page.
See PremiumBrowse the free reference

Related formulas

FTC Part 1

See all 19 Integrals formulas on one printable sheet - the same statements and conditions as these pages, typeset and laid out for paper.

Practice

Quiz yourself on Integrals

FTC Part 2 is one of 19 Integrals formulas in CalcRef, out of 136 in the library. Today's free round is 10 mixed multiple-choice questions drawn from the same dataset as this page, refreshed every day and replayable as often as you like - no account needed.

Take today's free quizAll 19 Integrals formulas