The definite integral
A definite integral has a lower limit a and an upper limit b. Its value is found by ∫ab f(x) dx = [F(x)]ab = F(b) − F(a), where F is any antiderivative of f. The constant c cancels, so it is omitted.
Key formula
∫ab f(x) dx = F(b) − F(a). Useful properties: ∫aa f dx = 0; ∫ab f dx = −∫ba f dx; ∫ab k·f dx = k ∫ab f dx; ∫ac f dx + ∫cb f dx = ∫ab f dx.
Worked example
Worked example
Evaluate ∫12 3x² dx. Antiderivative is x³, so [x³]12 = 2³ − 1³ = 8 − 1 = 7.
Using the properties
Example: ∫02 (2x + 1) dx = [x² + x]02 = (4 + 2) − 0 = 6. If ∫14 f dx = 5, then ∫14 3f dx = 15 by the constant-multiple rule.
Remember
- Substitute the upper limit first, then subtract the lower.
- No + c in a definite integral.
- Swapping the limits changes the sign.