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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.

Worked examples

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

Answer: 8

Evaluate ∫₀^π sin x dx.
  1. Antiderivative: F(x) = -cos x
  2. F(π) - F(0) = -cos π - (-cos 0) = -(-1) - (-1) = 1 + 1 = 2

Answer: 2

Related formulas

FTC Part 1

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