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Integrals · Fundamental Theorems

FTC Part 1

ddx[∫axf(t) dt]=f(x)\frac{d}{dx}\left[\int_a^x f(t)\, dt\right] = f(x)

The Fundamental Theorem of Calculus Part 1: The derivative of the integral (with variable upper limit) of a continuous function is the original function.

Conditions. f must be continuous on [a, x].

Worked examples

Find d/dx ∫₀ˣ sin(t²) dt.
  1. By FTC Part 1, the answer is simply sin(x²).

Answer: sin(x²)

Find d/dx ∫₁^(x²) √t dt.
  1. By FTC Part 1 with chain rule: √(x²) · d/dx[x²] = |x| · 2x = 2x|x|
  2. For x > 0 this simplifies to 2x²

Answer: 2x|x| (or 2x² for x > 0)

Related formulas

FTC Part 2

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