Antiderivatives and Basic Integration Rules
Simple Explanation
An antiderivative of f(x) is any function F(x) whose derivative is f(x). Integration is the process of finding antiderivatives, and it is exactly the reverse operation of differentiation — the Power Rule for integration, ∫xⁿdx = xⁿ⁺¹/(n+1) + C, undoes the Power Rule for differentiation.
Why Do We Need It?
Integration recovers a quantity from its rate of change — total distance from velocity, total cost from marginal cost, total accumulated growth from a growth rate — making it the natural counterpart to everything derivatives compute.
Formula
The Power Rule for Integration
∫xⁿ dx = xⁿ⁺¹/(n+1) + C (n ≠ −1)
Integration is the reverse of differentiation — this rule finds the family of functions whose derivative is xⁿ, up to an unknown constant C.
- n
- — the exponent on x (any real number except −1)
- C
- — the constant of integration — any constant has a derivative of 0, so it cannot be recovered from the derivative alone
When to use it: Whenever integrating a single power of x, or any polynomial (by applying the rule to each term).
Worked Example
Integrate a polynomial
Find ∫(3x² − 4x + 5) dx.
Why Does This Work?
Differentiating the answer must recover the original integrand — d/dx[x³−2x²+5x+C] = 3x²−4x+5, which is exactly f(x), confirming the antiderivative is correct. This "differentiate to check" step always verifies an integral.
Real-Life Example
Recovering total distance from a speed function
A car's speed at time t is known as a function v(t), but its total distance traveled is not directly recorded.
Integrating the speed function recovers the position function — total distance traveled is exactly the antiderivative of speed.
Practice
Evaluate the definite integral ∫₀¹ x² dx.
MediumCommon mistake
Forgetting the "+C" on an indefinite integral — since any constant's derivative is 0, infinitely many antiderivatives exist, differing only by a constant, and all of them must be represented by +C.
Quick Review
- ∫xⁿ dx = xⁿ⁺¹/(n+1) + C (n≠−1); integrate term by term for a polynomial.
- Always include "+C" for an indefinite integral.
- Check any integral by differentiating the answer — it should recover the original integrand.