The Fundamental Theorem of Calculus

You’ll be able to

State both parts of the Fundamental Theorem of Calculus
Differentiate accumulation functions, including with the chain rule
Evaluate definite integrals using antiderivatives

Two theorems, one deep idea

The Fundamental Theorem of Calculus (FTC) ties differentiation and integration together as inverse processes. Part 1 says that an accumulation function g(x) = ∫ₐˣ f(t) dt has derivative g′(x) = f(x): differentiating an integral with a variable upper limit just gives back the integrand. Part 2 (the evaluation theorem) says ∫ₐᵇ f(x) dx = F(b) − F(a), where F is any antiderivative of f. Together they turn area problems into antiderivative problems.

FTC Part 1 (with the chain rule)

d/dx ∫ₐ^{u(x)} f(t) dt = f(u(x)) · u′(x)

When the upper limit is a function u(x) rather than plain x, multiply by u′(x). This chain-rule twist is heavily tested.

FTC Part 2 (evaluation)

∫ₐᵇ f(x) dx = F(b) − F(a), where F′ = f

Any antiderivative works — the constant of integration cancels in the subtraction.

Accumulation functions

An accumulation function g(x) = ∫ₐˣ f(t) dt builds up net area under f from a fixed start a to a moving endpoint x. By FTC Part 1 its rate of growth is exactly f(x): g increases where f > 0, decreases where f < 0, and has a local extremum where f crosses zero. This reframes graph-analysis questions — the sign chart of f becomes the increasing/decreasing behavior of g, letting you reason about g without a formula.

Worked example

If g(x) = ∫₁^{x²} sin(t) dt, find g′(x).

1.The upper limit is u(x) = x², a function of x, so FTC Part 1 needs the chain rule.
2.Substitute the upper limit into the integrand: sin(u) = sin(x²).
3.Multiply by the derivative of the upper limit: u′(x) = 2x.
4.Combine: g′(x) = sin(x²) · 2x = 2x·sin(x²).

Answer: g′(x) = 2x·sin(x²). The extra factor 2x comes from the chain rule on the variable upper limit x².

Watch out

When the upper limit of an accumulation integral is a function of x, you must multiply by its derivative. Writing d/dx ∫₁^{x²} sin(t) dt = sin(x²) and forgetting the ·2x is a very common FTC error.

Checkpoint

If F(x) = ∫₀ˣ (3t² + 1) dt, what is F′(x)?

Checkpoint

Evaluate ∫₁³ 2x dx.

On the exam

FTC Part 1 questions on the AP exam almost always disguise a chain-rule step by using a non-trivial upper limit. Before answering, check whether the upper limit is plain x or a function of x — the latter needs the extra u′(x) factor.

Answer the 2 checkpoints as you read.