Integration Basics
Grade 12 · Math · Study Guide
Explore antiderivatives, definite integrals, and the Fundamental Theorem of Calculus.
Integration is the reverse of differentiation. An antiderivative of f(x) is any function F(x) such that F′(x) = f(x). The indefinite integral ∫ f(x) dx = F(x) + C, where C is the constant of integration.
The definite integral ∫ₐᵇ f(x) dx gives the net signed area between f and the x-axis from a to b. It is evaluated using the Fundamental Theorem of Calculus: ∫ₐᵇ f(x) dx = F(b) - F(a), where F is any antiderivative.
u-substitution is the most common integration technique and mirrors the chain rule. Set u = g(x), then du = g′(x) dx, transforming the integral into one easier to evaluate.
| Integral | Result |
|---|---|
| ∫ xⁿ dx (n ≠ −1) | xⁿ⁺¹/(n+1) + C |
| ∫ eˣ dx | eˣ + C |
| ∫ 1/x dx | ln|x| + C |
| ∫ cos x dx | sin x + C |
| ∫ sin x dx | −cos x + C |
| Part | Statement |
|---|---|
| Part 1 | d/dx[∫ₐˣ f(t) dt] = f(x) |
| Part 2 | ∫ₐᵇ f(x) dx = F(b) − F(a) |
Key terms
- Antiderivative
- A function F whose derivative equals f; F′(x) = f(x).
- Definite integral
- ∫ₐᵇ f(x) dx — the net signed area under f from a to b.
- Indefinite integral
- ∫ f(x) dx = F(x) + C — the general antiderivative.
- u-substitution
- Technique that reverses the chain rule by substituting u = g(x).
Practice questions
What is ∫ 4x³ dx?
- x⁴ + C
- 12x² + C
- 4x⁴ + C
- x⁴/3 + C
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Power rule: ∫ 4x³ dx = 4 · x⁴/4 + C = x⁴ + C.
Evaluate ∫₀² 3x² dx.
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Antiderivative: x³. FTC: [x³]₀² = 2³ - 0³ = 8.
Using u-substitution with u = x² + 1, what is ∫ 2x(x² + 1)⁴ dx?
- (x²+1)⁵/5 + C
- (x²+1)⁵ + C
- 2x(x²+1)⁵/5 + C
- (x²+1)⁴/4 + C
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du = 2x dx, so ∫ u⁴ du = u⁵/5 + C = (x²+1)⁵/5 + C.
∫ₐᵃ f(x) dx = 0 for any continuous function f.
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When the upper and lower limits are equal, F(a) - F(a) = 0, regardless of the function.
What is ∫ eˣ dx?
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eˣ is its own antiderivative: d/dx[eˣ] = eˣ, so ∫ eˣ dx = eˣ + C.
Fun facts
- The ∫ symbol is an elongated 'S' from the Latin 'summa' (sum), reflecting that integration is a continuous summing process.
- Integration can compute volumes, arc lengths, and surface areas — not just areas under curves.
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