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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.
  • 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

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Evaluate ∫₀^π sin x dx.

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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.

Common questions

What is the FTC Part 2?

The Fundamental Theorem of Calculus Part 2: A definite integral can be evaluated using any antiderivative F of f. It is one of the integrals formulas in the CalcRef reference.

When does the FTC Part 2 apply?

The FTC Part 2 holds under this condition: f must be continuous on [a, b]. F is any antiderivative of f.

How do you use the FTC Part 2?

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

Practice

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