The definite integral

Definition and notation

• The definite integral of f from a to b is written ∫_a^b f(x) dx, where a is the lower limit and b is the upper limit.
• Fundamental theorem: if F is any antiderivative of f, then ∫_a^b f(x) dx = F(b) − F(a).
• The constant of integration cancels: (F(b) + C) − (F(a) + C) = F(b) − F(a). So never write + C in a definite integral.

Worked examples

1. ∫₀² (3x² + 2x) dx = [x³ + x²]₀² = (8 + 4) − 0 = 12
2. ∫₁⁴ x^−1/2 dx = [2√x]₁⁴ = 4 − 2 = 2
3. ∫₀^π/2 cos x dx = [sin x]₀^π/2 = 1 − 0 = 1
4. ∫₀¹ e^2x dx = [½e^2x]₀¹ = ½(e² − 1) ≈ 3.19

Integration by substitution in a definite integral

• Either change the limits to the new variable, or substitute back before applying the original limits.
• Example: ∫₀¹ 2x(x² + 1)³ dx. Let u = x² + 1, du = 2x dx; limits become 1 and 2: ∫₁² u³ du = [u⁴/4]₁² = 4 − ¼ = 15/4.