Lesson 1: Antiderivatives and Indefinite Integrals
Estimated time: 35-45 minutes
Learning Objectives
- Define antiderivative and indefinite integral
- Apply basic integration rules (power rule, trig, exponential)
- Understand the constant of integration +C
- Solve initial value problems
What Is an Antiderivative?
Differentiation takes a function and finds its rate of change. Antidifferentiation reverses this: given a rate of change, find the original function.
Antiderivative
F is an antiderivative of f if F'(x) = f(x). If F is one antiderivative, then F(x) + C represents all antiderivatives (C is an arbitrary constant).
Indefinite Integral Notation
∫ f(x) dx = F(x) + C
The ∫ symbol means "antiderivative of." The dx tells us the variable of integration.
Basic Integration Rules
Power Rule for Integration
∫ xn dx = xn+1/(n+1) + C, for n ≠ −1
Other Essential Rules
∫ 1/x dx = ln|x| + C ∫ ex dx = ex + C
∫ sin x dx = −cos x + C ∫ cos x dx = sin x + C
∫ sec² x dx = tan x + C ∫ csc² x dx = −cot x + C
Example 1: Power Rule
∫ (3x² + 5x − 2) dx = x³ + (5/2)x² − 2x + C.
Example 2: Trig and Exponential
∫ (2 cos x + ex) dx = 2 sin x + ex + C.
Example 3: Rewriting Before Integrating
∫ (x² + 1)/x dx = ∫ (x + 1/x) dx = x²/2 + ln|x| + C.
Initial Value Problems
An initial value problem gives you f'(x) and a specific value f(a) = b. You integrate, then use the condition to find C.
Example 4: Finding a Particular Antiderivative
Given f'(x) = 6x − 4 and f(1) = 3. Find f(x).
Step 1: f(x) = 3x² − 4x + C.
Step 2: f(1) = 3 − 4 + C = 3, so C = 4.
Answer: f(x) = 3x² − 4x + 4.
Key Takeaways
- An antiderivative reverses differentiation: if F' = f, then F is an antiderivative of f.
- The indefinite integral ∫ f(x) dx = F(x) + C includes the constant of integration.
- The power rule for integration: add 1 to the exponent, divide by the new exponent.
- For initial value problems, use the given condition to find C.
Check Your Understanding
1. Find ∫ (4x³ − 2x + 7) dx.
2. Find ∫ sec² x dx.
3. Find ∫ (3/x + 2ex) dx.
4. If f'(x) = cos x and f(0) = 2, find f(x).