When F(x) is an antiderivative of f(x),
∫_a^b f(x) dx = F(b) − F(a) (definite integral)
Basic Properties:
- ∫_a^a f dx = 0
- ∫_a^b f dx = −∫_b^a f dx
- ∫_a^c f dx = ∫_a^b f dx + ∫_b^c f dx
- ∫_a^b (f + g) dx = ∫_a^b f dx + ∫_a^b g dx
Example ① Calculate ∫_0^2 (3x²−2x) dx.
Solution:The antiderivative is F(x) = x³ − x².
F(2) − F(0) = (8−4) − 0 = 4
Solution:The antiderivative is F(x) = x³ − x².
F(2) − F(0) = (8−4) − 0 = 4
Key Point
- A definite integral is a real number (an indefinite integral is a function+C)
- Notation: [F(x)]_a^b = F(b) − F(a)
Practice Problems
- ∫_1^3 (2x+1) dx
- ∫_0^{π/2} cos x dx
Answer and Explanation
- Answer:10
[x²+x]_1^3 = 12 − 2 = 10. - Answer:1
[sin x]_0^{π/2} = 1 − 0.