Antidifferentiation Rules
Reading Differentiation Backwards
Every differentiation rule, when read in reverse, gives you an antidifferentiation rule. Know that $\frac{d}{dx}(x^3) = 3x^2$? Then you also know an antiderivative of $3x^2$ is $x^3$.
The key formulas in this section let you antidifferentiate polynomials, rational functions (with integer powers), and basic trig functions instantly, without computing any limits.
Think of it as building a vocabulary. The more differentiation formulas you’ve memorized, the more antiderivatives you can recognize on sight.
Prerequisite Map
Quick Reference
| Property | Value |
|---|---|
| Concept | Antiderivatives |
| Chapter | 3.9 |
| Difficulty | Beginner |
| Time | ~20 minutes |
Key Concepts
The Power Rule for Antiderivatives
The derivative rule $\frac{d}{dx}(x^n) = nx^{n-1}$ reverses to:
$$\boxed{\int x^n\, dx = \frac{x^{n+1}}{n+1} + C \quad (n \neq -1)}$$
In words: “Add one to the power, divide by the new power.”
Why does this work? Check by differentiating: $$\frac{d}{dx}\left(\frac{x^{n+1}}{n+1}\right) = \frac{(n+1)x^n}{n+1} = x^n \checkmark$$
Warning: This formula fails when $n = -1$ because you’d be dividing by zero. The antiderivative of $x^{-1} = 1/x$ is $\ln\vert x\vert + C$, which you’ll learn later.
Constant Multiple Rule
$$\int cf(x)\,dx = c\int f(x)\,dx$$
Constants “pull out” of the antiderivative, just like with derivatives.
Sum and Difference Rules
$$\int [f(x) + g(x)]\,dx = \int f(x)\,dx + \int g(x)\,dx$$
$$\int [f(x) - g(x)]\,dx = \int f(x)\,dx - \int g(x)\,dx$$
You can antidifferentiate term by term.
Table of Basic Antiderivatives
| Function $f(x)$ | Antiderivative $F(x)$ | Notes |
|---|---|---|
| $x^n$ $(n \neq -1)$ | $\frac{x^{n+1}}{n+1} + C$ | Power rule |
| $\sin x$ | $-\cos x + C$ | Derivative of $-\cos x$ is $\sin x$ |
| $\cos x$ | $\sin x + C$ | Derivative of $\sin x$ is $\cos x$ |
| $\sec^2 x$ | $\tan x + C$ | Derivative of $\tan x$ is $\sec^2 x$ |
| $\sec x \tan x$ | $\sec x + C$ | Derivative of $\sec x$ is $\sec x \tan x$ |
| $\csc^2 x$ | $-\cot x + C$ | Derivative of $-\cot x$ is $\csc^2 x$ |
| $\csc x \cot x$ | $-\csc x + C$ | Derivative of $-\csc x$ is $\csc x \cot x$ |
Working with Negative and Fractional Exponents
Convert to exponential form before applying the power rule:
| Expression | Rewrite as | Antiderivative |
|---|---|---|
| $\frac{1}{x^3}$ | $x^{-3}$ | $\frac{x^{-2}}{-2} + C = -\frac{1}{2x^2} + C$ |
| $\sqrt{x}$ | $x^{1/2}$ | $\frac{x^{3/2}}{3/2} + C = \frac{2}{3}x^{3/2} + C$ |
| $\frac{1}{\sqrt{x}}$ | $x^{-1/2}$ | $\frac{x^{1/2}}{1/2} + C = 2\sqrt{x} + C$ |
| $\sqrt[3]{x^2}$ | $x^{2/3}$ | $\frac{x^{5/3}}{5/3} + C = \frac{3}{5}x^{5/3} + C$ |
Combining Rules: Polynomials
For a polynomial, antidifferentiate term by term:
$$\int (4x^3 - 5x^2 + 2x - 7)\,dx = x^4 - \frac{5x^3}{3} + x^2 - 7x + C$$
Key: Only ONE constant $C$ at the end, not one per term!
Practice Problems
Find the general antiderivative of $f(x) = x^5$.
Find the general antiderivative of $f(x) = \sqrt{x}$.
Find the general antiderivative of $g(x) = 6x^2 - 4x + 9$.
Find the general antiderivative of
$$h(x) = \frac{3x^4 - 2\sqrt{x}}{x}$$
Find all functions $f$ such that
$$f'(x) = 5\sin x + \frac{3}{x^4} - 2\sec^2 x + 7$$
CCI-Style Conceptual Questions
Without computing, predict: is the antiderivative of $x^{99}$ a polynomial of degree 98, 99, or 100?
Explain in your own words why the formula $\int x^n\,dx = \frac{x^{n+1}}{n+1} + C$ cannot be used when $n = -1$.
True or False: If $\frac{d}{dx}[F(x)] = f(x)$, then $\frac{d}{dx}[f(x)] = F(x)$.
Common Misconceptions
the antiderivative of a product is the product of the antiderivatives.
This is the multiplicative-not-additive error. No product rule exists for antiderivatives; $\int f(x)g(x)\,dx$ is generally not equal to $\left(\int f(x)\,dx\right)\left(\int g(x)\,dx\right)$. A direct counterexample: $\int x \cdot x\,dx = \int x^2\,dx = \frac{x^3}{3} + C$, but $\left(\int x\,dx\right)\left(\int x\,dx\right) = \frac{x^2}{2} \cdot \frac{x^2}{2} = \frac{x^4}{4}$, which is a different function. The sum and constant-multiple rules do extend from differentiation to antidifferentiation, but the product rule does not.
each term in a sum requires its own arbitrary constant $C$.
This is the concept-image-conflicts-definition error about the structure of the general antiderivative. When antidifferentiating a sum term by term, each term produces its own constant, but all of those constants combine into a single arbitrary constant $C$ in the final answer. Writing $x^4 + C_1 - \frac{5x^3}{3} + C_2 + x^2 + C_3 - 7x + C_4$ is redundant; $C_1 + C_2 + C_3 + C_4$ is still an arbitrary constant, so the answer is correctly written as $x^4 - \frac{5x^3}{3} + x^2 - 7x + C$.
Mastery Checklist
Mental Model
Antidifferentiation is “undoing” differentiation:
| Differentiation (Power Rule) | Antidifferentiation (Reverse Power Rule) |
|---|---|
| Multiply by the power | Divide by the new power |
| Subtract 1 from the power | Add 1 to the power |
Think of it as watching a movie in reverse: every step of differentiation has a corresponding “undo” step in antidifferentiation.
But remember: antidifferentiation loses information (the “+C”), so you can’t fully recover the original function without additional information.
| Previous | Up | Next |
|---|---|---|
| Antiderivative Definition | Skills Index | Initial Value Problems |
Last updated: 2026-01-22