A function F(x) satisfying F'(x) = f(x) is called an antiderivative of f(x), and
∫ f(x) dx = F(x) + C （C is the constant of integration）is written. This is the indefinite integral。
Basic formulas：
- ∫ x^n dx = x^{n+1}/(n+1) + C （n ≠ −1）
- ∫ 1/x dx = log|x| + C
- ∫ e^x dx = e^x + C
- ∫ sin x dx = −cos x + C、∫ cos x dx = sin x + C
- ∫ 1/cos²x dx = tan x + C
Example ① Calculate the following.
(1) ∫ (3x²−4x+1) dx (2) ∫ √x dx (3) ∫ (2/x + e^x) dx
Solution:
(1) x³ − 2x² + x + C
(2) ∫ x^{1/2} dx = (2/3)x^{3/2} + C = (2/3)x√x + C
(3) 2 log|x| + e^x + C
(1) ∫ (3x²−4x+1) dx (2) ∫ √x dx (3) ∫ (2/x + e^x) dx
Solution:
(1) x³ − 2x² + x + C
(2) ∫ x^{1/2} dx = (2/3)x^{3/2} + C = (2/3)x√x + C
(3) 2 log|x| + e^x + C
Key Point
- Integration is the inverse operation of differentiation
- Always write +C (the constant of integration)
- When n = −1 in x^n, use log|x|
Practice Problems
- ∫ (4x³ − 6x + 2) dx
- ∫ (3sin x − 2cos x) dx
Solutions and Explanations
- Solution:x⁴ − 3x² + 2x + C
- Solution:−3cos x − 2sin x + C