Fundamental Theorem of Calculus Formula

Fundamental theorem of calculus are the theorem stating that differentiation and integration are inverse operations, linking antiderivatives to definite.

The Formula

Part 1: ddx∫axf(t) dt=f(x)\frac{d}{dx}\int_a^x f(t)\,dt = f(x). Part 2: ∫abf(x) dx=F(b)−F(a)\int_a^b f(x)\,dx = F(b) - F(a) where F′=fF' = f.

When to use: Integration undoes differentiation. They're two sides of the same coin.

Quick Example

ddx[∫0xf(t) dt]=f(x)\frac{d}{dx}\left[\int_0^x f(t) \, dt\right] = f(x) The derivative of the integral is the original function.

Notation

FTC Part 1 and FTC Part 2. FF denotes any antiderivative of ff, i.e., F′(x)=f(x)F'(x) = f(x).

What This Formula Means

The theorem stating that differentiation and integration are inverse operations, linking antiderivatives to definite integrals.

Integration undoes differentiation. They're two sides of the same coin.

Formal View

Part 1: If ff is continuous on [a,b][a, b] and G(x)=∫axf(t) dtG(x) = \int_a^x f(t)\,dt, then G′(x)=f(x)G'(x) = f(x) for all x∈(a,b)x \in (a, b). Part 2: If ff is continuous on [a,b][a, b] and F′=fF' = f, then ∫abf(x) dx=F(b)−F(a)\int_a^b f(x)\,dx = F(b) - F(a).

Worked Examples

Example 1

easy
Let G(x)=∫0x(t2+1) dtG(x) = \int_0^x (t^2 + 1)\,dt. Find G′(x)G'(x) using FTC Part 1.

Answer

G′(x)=x2+1G'(x) = x^2 + 1

First step

1
FTC Part 1 states: if G(x)=∫axf(t) dtG(x) = \int_a^x f(t)\,dt, then G′(x)=f(x)G'(x) = f(x).

Full solution

  1. 2
    Here f(t)=t2+1f(t) = t^2 + 1, so G′(x)=f(x)=x2+1G'(x) = f(x) = x^2 + 1.
  2. 3
    No integration is needed — the derivative of an integral with variable upper limit is just the integrand evaluated at xx.
FTC Part 1 says differentiation undoes integration when the upper limit is the variable. You simply replace tt with xx in the integrand. This is the key link showing derivatives and integrals are inverse operations.

Example 2

hard
Let H(x)=∫1x2cos⁡t dtH(x) = \int_1^{x^2} \cos t\,dt. Find H′(x)H'(x).

Common Mistakes

  • Mixing up the parts — Part 1 is about differentiating an integral, Part 2 is about evaluating one with an antiderivative.
  • Forgetting the chain-rule factor in Part 1 when the upper bound is a function — ddx∫ag(x)f dt=f(g(x))g′(x)\frac{d}{dx}\int_a^{g(x)}f\,dt=f(g(x))g'(x).
  • Plugging the lower bound into the Part-1 derivative — the result is just ff evaluated at the variable upper bound, with no −f(a)-f(a) term.

Why This Formula Matters

Before this theorem, integrals meant slow limits of Riemann sums; FTC makes them computable with antiderivatives, which is why calculus is usable at all. The deeper idea is that the accumulated area function carries the original rate inside it — its slope at xx is exactly f(x)f(x) — tying the whole subject into one loop. Recognizing it by "Am I connecting a definite integral to an antiderivative (F(b)−F(a)F(b)-F(a)) or differentiating an accumulation function back to its integrand?" — rather than by familiar numbers — is what lets a student tell it apart from riemann sums and chain rule (with ftc part 1) and definite integral (mechanics) in a mixed problem set.

Frequently Asked Questions

What is the Fundamental Theorem of Calculus formula?

The theorem stating that differentiation and integration are inverse operations, linking antiderivatives to definite integrals.

How do you use the Fundamental Theorem of Calculus formula?

Integration undoes differentiation. They're two sides of the same coin.

What do the symbols mean in the Fundamental Theorem of Calculus formula?

FTC Part 1 and FTC Part 2. FF denotes any antiderivative of ff, i.e., F′(x)=f(x)F'(x) = f(x).

Why is the Fundamental Theorem of Calculus formula important in Math?

Before this theorem, integrals meant slow limits of Riemann sums; FTC makes them computable with antiderivatives, which is why calculus is usable at all. The deeper idea is that the accumulated area function carries the original rate inside it — its slope at xx is exactly f(x)f(x) — tying the whole subject into one loop. Recognizing it by "Am I connecting a definite integral to an antiderivative (F(b)−F(a)F(b)-F(a)) or differentiating an accumulation function back to its integrand?" — rather than by familiar numbers — is what lets a student tell it apart from riemann sums and chain rule (with ftc part 1) and definite integral (mechanics) in a mixed problem set.

What do students get wrong about Fundamental Theorem of Calculus?

The procedure for fundamental theorem of calculus is the easy part; the trap is mixing up the parts. Asking "Am I connecting a definite integral to an antiderivative (F(b)−F(a)F(b)-F(a)) or differentiating an accumulation function back to its integrand?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

What should I learn before the Fundamental Theorem of Calculus formula?

Before studying the Fundamental Theorem of Calculus formula, you should understand: derivative, integral.

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Related Concepts

DerivativeIntegral