The indefinite integral is the calculus tool used to reverse differentiation. If a derivative tells you a rate of change, an indefinite integral helps recover the original family of functions that could have produced that rate. It matters because many problems in physics, engineering, economics, and geometry begin with a known rate and ask for the accumulated quantity.
The symbol ∫ f(x) dx means to find all antiderivatives of f(x).
The key idea is that if F'(x) = f(x), then ∫ f(x) dx = F(x) + C. The added constant C appears because many functions that differ only by a constant have the same derivative. Basic integration rules let you build antiderivatives for powers, sums, constant multiples, exponentials, and trigonometric functions.
Checking an indefinite integral is simple: differentiate your answer and see whether you get the original integrand.
Key Facts
- Main definition: ∫ f(x) dx = F(x) + C, where F'(x) = f(x).
- Power rule: ∫ x^n dx = x^(n + 1)/(n + 1) + C, for n ≠ -1.
- Special case: ∫ 1/x dx = ln|x| + C.
- Constant multiple rule: ∫ k f(x) dx = k∫ f(x) dx.
- Sum rule: ∫ [f(x) + g(x)] dx = ∫ f(x) dx + ∫ g(x) dx.
- Check rule: If d/dx [F(x) + C] = f(x), then F(x) + C is a correct indefinite integral.
Vocabulary
- Indefinite integral
- An expression that represents the full family of antiderivatives of a function.
- Antiderivative
- A function F(x) whose derivative is the given function f(x).
- Constant of integration
- The arbitrary constant C added to an indefinite integral because derivatives of constants are zero.
- Integrand
- The function being integrated, such as f(x) in ∫ f(x) dx.
- Differential
- The dx in an integral that identifies the variable of integration.
Common Mistakes to Avoid
- Forgetting + C: This is wrong because an indefinite integral represents a family of functions, not just one function.
- Using the power rule on x^-1: This is wrong because the power rule formula would divide by zero, so ∫ 1/x dx = ln|x| + C instead.
- Changing the variable accidentally: This is wrong because ∫ f(x) dx must be integrated with respect to x, while ∫ f(t) dt is a different notation choice.
- Not checking by differentiating: This can hide algebra errors because the derivative of your answer must reproduce the original integrand exactly.
Practice Questions
- 1 Find ∫ (6x^2 - 4x + 9) dx.
- 2 Find ∫ (3/x + 5e^x - 2cos x) dx.
- 3 Two antiderivatives of the same function are F(x) = x^3 + 2x + 5 and G(x) = x^3 + 2x - 7. Explain why they have the same derivative and what this shows about + C.