What you need to know

Integration is the reverse of differentiation. Differentiating x³ gives 3x², so integrating 3x² gives back x³ — almost. Because the derivative of any constant is 0, integrating 3x² could give x³ + 1 or x³ − 7 just as well. We write the answer as x³ + C, where C is the constant of integration. The integral sign is ∫, and ∫ f(x) dx means "integrate f(x) with respect to x".

The power rule for integration. ∫ x[sup n] dx = [frac x[sup n + 1]/n + 1] + C (for n ≠ −1). Raise the power by 1, then divide by the new power. Constant multipliers stay in front, and sums are integrated term by term.

Definite integrals and area. A definite integral has limits: ∫ from a to b of f(x) dx. Integrate, then substitute the upper limit and subtract the value at the lower limit: F(b) − F(a). The constant C cancels, so it is left out. When f(x) ≥ 0 between a and b, the definite integral gives the area between the curve and the x-axis.

Check yourself

1. ∫ x⁴ dx
2. ∫ (6x² − 4x + 5) dx
3. ∫ 1x³ dx
4. f′(x) = 4x and f(2) = 10. Find f(x).
5. ∫ from 0 to 2 of x³ dx
6. Find the area under y = 2x + 1 from x = 1 to x = 3.

Answers: 1. x⁵5 + C 2. 2x³ − 2x² + 5x + C 3. −12x² + C (that is, x⁻²−2 + C) 4. f(x) = 2x² + C with 8 + C = 10, so f(x) = 2x² + 2 5. 164 − 0 = 4 6. F(x) = x² + x; F(3) − F(1) = 12 − 2 = 10 square units