Step by Step
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Part 1 — differentiating an integral
If you define a function as the integral of f from a fixed point a up to a variable x, then differentiating that function just gives you back f(x): d/dx[∫ₐˣ f(t)dt] = f(x). Integration and differentiation undo each other.
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Part 2 — evaluating a definite integral
To evaluate ∫ₐᵇ f(x)dx, find any antiderivative F of f, then compute F(b) − F(a). This turns the (often hard) process of computing an area under a curve into simple substitution once you have an antiderivative.
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Why this theorem is such a big deal
Before this connection was established, computing areas under curves (integration) and computing rates of change (differentiation) were treated as two completely separate problems. The Fundamental Theorem proves they are, in fact, inverse operations of each other.
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Any antiderivative works for Part 2
When evaluating F(b) − F(a), it doesn't matter which specific antiderivative F you pick (they differ only by a constant) — the constant of integration always cancels out in the subtraction.
Applied Walkthrough
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Evaluate ∫₁³ 2x dx using the Fundamental Theorem of Calculus, Part 2.
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Find an antiderivative of 2x: F(x) = x² (choosing the simplest antiderivative, with the constant of integration set to 0, since it will cancel anyway).
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Apply Part 2: ∫₁³ 2x dx = F(3) − F(1) = 3² − 1² = 9 − 1 = 8.
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This shows how the theorem converts a definite integral (a geometric area calculation) into a simple two-step arithmetic evaluation, once an antiderivative is known.
Exam Application
Exams test whether you understand both parts of the Fundamental Theorem — that differentiating an integral gives back the original function (Part 1), and that a definite integral can be evaluated via F(b) − F(a) using any antiderivative (Part 2).
⚠ Common Trap
The most common trap is forgetting that any valid antiderivative can be used in Part 2 — students sometimes worry about which specific antiderivative to pick, but the constant of integration always cancels out in the subtraction F(b) − F(a).
✓ Quick Self-Check
1. What does Part 1 of the Fundamental Theorem of Calculus state?
d/dx[∫ₐˣ f(t)dt] = f(x) — differentiating an integral gives back the original function.
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2. What does Part 2 of the Fundamental Theorem of Calculus state?
∫ₐᵇ f(x)dx = F(b) − F(a), where F is any antiderivative of f.
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3. Evaluate ∫₁³ 2x dx.
8.
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4. Why doesn't it matter which specific antiderivative you use in Part 2?
Because different antiderivatives only differ by a constant, and that constant cancels out when computing F(b) − F(a).
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5. What relationship does the Fundamental Theorem of Calculus establish between differentiation and integration?
That they are inverse operations of one another.
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