Integration is the inverse operation of differentiation. An arbitrary constant C is required in an indefinite integral.
Definition of an Indefinite Integral
When F'(x) = f(x), ∫f(x)dx = F(x) + C（C is the constant of integration）
Basic Formulas
- ∫xⁿ dx = xⁿ⁺¹/(n+1) + C（n ≠ −1）
- ∫k dx = kx + C（k is a constant）
- ∫(f+g) dx = ∫f dx + ∫g dx
- ∫kf dx = k∫f dx
Example ①
Integrate the following. (1) ∫(3x² − 2x + 1)dx (2) ∫(x+1)² dx
Answer:
(1) x³ − x² + x + C
(2) (x+1)² = x²+2x+1 → x³/3 + x² + x + C
Integrate the following. (1) ∫(3x² − 2x + 1)dx (2) ∫(x+1)² dx
Answer:
(1) x³ − x² + x + C
(2) (x+1)² = x²+2x+1 → x³/3 + x² + x + C
Example ②
Find f(x) when f'(x) = 3x² − 2 and f(0) = 4.
Answer:f(x) = x³ − 2x + C。f(0) = C = 4 → f(x) = x³ − 2x + 4
Find f(x) when f'(x) = 3x² − 2 and f(0) = 4.
Answer:f(x) = x³ − 2x + C。f(0) = C = 4 → f(x) = x³ − 2x + 4
Key Points
- Integration is “reverse differentiation”: find a function whose derivative is f(x)
- ∫xⁿ dx: add 1 to the exponent, then divide by the resulting number
- Always write the constant of integration C
Practice Problems
- Integrate the following: (1) ∫(4x³ − 6x + 2)dx (2) ∫(x²−1)²dx
- Find f(x) when f'(x) = 6x² − 4x and f(1) = 3.
- Calculate ∫(2x+1)(x−3)dx.
Answers and Explanations
- Answer:(1) x⁴−3x²+2x+C (2) x⁵/5−2x³/3+x+C
Explanation: Expand (x²−1)²=x⁴−2x²+1 in (2), then integrate. - Answer:f(x) = 2x³−2x²+3
Explanation: f(x)=2x³−2x²+C。f(1)=2−2+C=3→C=3。 - Answer:2x³/3 − 5x²/2 − 3x + C
Explanation: (2x+1)(x−3)=2x²−5x−3。∫(2x²−5x−3)dx。