Indefinite integrals
Integration reverses differentiation. If F′(x) = f(x), then ∫f(x) dx = F(x) + C, where C is the constant of integration. Standard results:
- ∫xⁿ dx = xⁿ⁺¹/(n + 1) + C (n ≠ −1)
- ∫(1/x) dx = log|x| + C
- ∫eˣ dx = eˣ + C; ∫aˣ dx = aˣ/log a + C
- ∫k f(x) dx = k∫f(x) dx; the integral of a sum is the sum of integrals.
Example: ∫(4x³ − 6x) dx = x⁴ − 3x² + C.
Definite integrals
Fundamental theorem: ∫ₐᵇ f(x) dx = F(b) − F(a), where F is any antiderivative. No constant is needed. Example: ∫₀² (2x + 1) dx = [x² + x]₀² = 6. Similarly, ∫₀¹ eˣ dx = e − 1.
Area under a curve
If f(x) ≥ 0 on [a, b], the area between the curve y = f(x), the x-axis and the lines x = a and x = b is ∫ₐᵇ f(x) dx. If the curve lies below the axis, take the absolute value of the integral.
Example: the parabola y = 4 − x² meets the x-axis at x = ±2. The area it encloses with the x-axis is ∫₋₂² (4 − x²) dx = 16 − 16/3 = 32/3 square units.
Area between two curves
If f(x) ≥ g(x) on [a, b], the area between them is ∫ₐᵇ [f(x) − g(x)] dx. The limits are usually where the curves meet. Example: y = x and y = x² meet at x = 0 and x = 1, and x ≥ x² there, so the area is ∫₀¹ (x − x²) dx = 1/2 − 1/3 = 1/6.
Economic meaning
Integration recovers a total from a marginal function: if MC is known, total cost C(x) = ∫MC dx + fixed cost.
Key points
- Indefinite integral: family of functions + C.
- Definite integral: a number, F(b) − F(a).
- Area is always taken as positive.
- Area between curves: integrate (upper − lower).
Exam tips
- Sketch the curves to find the limits and to see which curve is on top.
- Split the interval where a curve crosses the x-axis.