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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].
  • 2 worked examples
  • 19 in Integrals

Worked examples

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

Answer: sin(x²)

1 more worked examplePremium

Find d/dx ∫₁^(x²) √t dt.

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Related formulas

FTC Part 2

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 1?

The Fundamental Theorem of Calculus Part 1: The derivative of the integral (with variable upper limit) of a continuous function is the original function. It is one of the integrals formulas in the CalcRef reference.

When does the FTC Part 1 apply?

The FTC Part 1 holds under this condition: f must be continuous on [a, x].

How do you use the FTC Part 1?

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

Practice

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