# Antiderivatives and indefinite integrals (basic rules)

> AP Calculus BC · CED — Integration and Accumulation of Change
> Source: https://www.owlsprep.com/study/ap-calculus-bc-u6-antiderivatives-and-indefinite-integrals/

This guide covers core definitions of antiderivatives and indefinite integrals, basic integration rules for elementary functions, common exam traps, and worked examples aligned to AP Calculus BC expectations.

**Prerequisites:** Derivative rules for all elementary functions; Algebraic manipulation of polynomials and rational functions; Basic function notation for derivatives

## Learning objectives

- Define antiderivatives and indefinite integrals with correct notation
- Apply basic antiderivative rules to algebraic, exponential, logarithmic and trigonometric functions
- Identify and avoid common integration errors tested on the AP exam
- Use basic antiderivatives to solve initial value and applied motion problems

## Core Definitions and Notation

An antiderivative of a function $f(x)$ is a differentiable function $F(x)$ such that $F'(x) = f(x)$ for all $x$ in the domain of $f$. The indefinite integral of $f(x)$ is the general form of all antiderivatives of $f$, which form a family of functions differing only by a vertical shift.

**Indefinite Integral** — The general antiderivative of $f(x)$, which includes an arbitrary constant of integration $C$ to represent all possible antiderivatives of $f$.

*Notation:* $\int f(x) dx$

*Example:* $\int 2x dx = x^2 + C$, where all vertical shifts of $x^2$ are antiderivatives of $2x$.

This topic is the foundational starting point for all integration work in AP Calculus BC, making up 1-3% of total exam score. It acts as a building block for all larger problems including differential equations, area/volume calculations, and accumulation of change.

> **info**
>
> Unlike definite integrals, which evaluate to a numerical quantity representing net change, indefinite integrals describe an entire family of functions.

## Core Algebraic Antiderivative Rules

All basic antiderivative rules are derived by reversing corresponding derivative rules. The constant multiple and sum/difference rules allow us to pull constants out of integrals and integrate term-by-term:

$$\int k f(x) dx = k \int f(x) dx$$

$$\int \left(f(x) \pm g(x)\right) dx = \int f(x) dx \pm \int g(x) dx$$

The power rule for integration reverses the power rule for derivatives, with a critical exception for $n=-1$:

$$\int x^n dx = \frac{x^{n+1}}{n+1} + C, \quad n \neq -1$$

When $n=-1$, the integrand is $\frac{1}{x}$, and the power rule would result in division by zero. For this case, the antiderivative is $\ln|x| + C$, since the derivative of $\ln|x|$ is $\frac{1}{x}$ for all non-zero $x$.

**Worked example:** Find the general indefinite integral $\int \left(4x^3 - \frac{2}{x^2} + 5\right) dx$.

1. First, rewrite all terms with exponents to fit the power rule:
2. $$\frac{2}{x^2} = 2x^{-2}, \quad 5 = 5x^0$$
3. Apply sum/difference and constant multiple rules to split the integral:
4. $$\int 4x^3 dx - \int 2x^{-2} dx + \int 5x^0 dx = 4\int x^3 dx - 2\int x^{-2} dx + 5\int x^0 dx$$
5. Apply the power rule to each term: First term: $4 \cdot \frac{x^{3+1}}{3+1} = 4 \cdot \frac{x^4}{4} = x^4$. Second term: $-2 \cdot \frac{x^{-2+1}}{-2+1} = -2 \cdot \frac{x^{-1}}{-1} = 2x^{-1} = \frac{2}{x}$. Third term: $5 \cdot \frac{x^{0+1}}{0+1} = 5x$.
6. Add the arbitrary constant of integration $C$ to get the final result:
7. $$\int \left(4x^3 - \frac{2}{x^2} + 5\right) dx = x^4 + \frac{2}{x} + 5x + C$$

> **Exam tip:** Always rewrite roots and rational terms as $x^n$ before applying the power rule to avoid mistakes with negative and fractional exponents.

## Exponential and Logarithmic Antiderivative Rules

Reversing derivative rules for exponential and logarithmic functions gives core rules frequently used in applied calculus problems. The simplest rule is for the natural exponential function, which is its own antiderivative:

$$\int e^x dx = e^x + C$$

For general exponential functions with constant base $a \neq e$, we reverse the derivative rule $\frac{d}{dx}[a^x] = a^x \ln a$ by dividing by $\ln a$:

$$\int a^x dx = \frac{a^x}{\ln a} + C, \quad a>0, a \neq 1$$

For the natural logarithm itself, the antiderivative is derived from the product rule for derivatives:

$$\int \ln x dx = x \ln x - x + C, \quad x>0$$

> **warning**
>
> A common mistake is mixing up power functions ($x^n$, variable base, constant exponent) and exponential functions ($a^x$, constant base, variable exponent), leading to misapplying the wrong rule.

**Worked example:** Evaluate the indefinite integral $\int \left(3e^x + 2^x - \frac{4}{x}\right) dx$.

1. Split the integral using the sum/difference rule:
2. $$3\int e^x dx + \int 2^x dx - 4 \int \frac{1}{x} dx$$
3. Apply the corresponding rule to each term: $\int e^x dx = e^x$, $\int 2^x dx = \frac{2^x}{\ln 2}$, $\int \frac{1}{x} dx = \ln|x|$.
4. Substitute back, multiply by constants, and add $C$ to get the final result:
5. $$\int \left(3e^x + 2^x - \frac{4}{x}\right) dx = 3e^x + \frac{2^x}{\ln 2} - 4 \ln|x| + C$$

> **Exam tip:** Always check if your function is a power function or exponential function before integrating; if the variable is in the exponent, use the exponential rule, not the power rule.

## Trigonometric Antiderivative Rules

All basic trigonometric antiderivative rules are derived directly from reversing derivative rules, and these are not provided on the AP Calculus formula sheet, so memorization is required:

- $\int \cos x dx = \sin x + C$
- $\int \sin x dx = -\cos x + C$
- $\int \sec^2 x dx = \tan x + C$
- $\int \csc^2 x dx = -\cot x + C$
- $\int \sec x \tan x dx = \sec x + C$
- $\int \csc x \cot x dx = -\csc x + C$

> **tip**
>
> Negative signs are the most common source of error here. A 10-second check by differentiating your final result confirms it matches the original integrand.

**Worked example:** Find the general antiderivative of $f(x) = 2 \sin x - \sec^2 x + 3 \sec x \tan x$.

1. Split into separate integrals using the sum rule:
2. $$2 \int \sin x dx - \int \sec^2 x dx + 3 \int \sec x \tan x dx$$
3. Apply trigonometric antiderivative rules to each term: $2 \int \sin x dx = 2(-\cos x) = -2 \cos x$, $- \int \sec^2 x dx = - \tan x$, $3 \int \sec x \tan x dx = 3 \sec x$.
4. Combine terms and add the arbitrary constant $C$ to get the final result:
5. $$\int (2 \sin x - \sec^2 x + 3 \sec x \tan x) dx = -2 \cos x - \tan x + 3 \sec x + C$$

**Check your understanding**

Test your understanding with this AP-style multiple choice question:

1. Which of the following is the general antiderivative of $f(x) = 2\sqrt{x} - \frac{3}{\sqrt{x}} + 5\cos x$?

   - $\frac{4}{3}x^{3/2} - 6\sqrt{x} + 5 \sin x + C$
   - $\frac{4}{3}x^{3/2} - \frac{3}{2}\sqrt{x} + 5 \sin x + C$
   - $2x^{3/2} - 3x^{1/2} + 5 \sin x + C$
   - $\frac{4}{3}x^{3/2} - 6\sqrt{x} - 5 \sin x + C$

   *Answer:* $\frac{4}{3}x^{3/2} - 6\sqrt{x} + 5 \sin x + C$

   *Why:* Correct! Rewriting $2\sqrt{x} = 2x^{1/2}$ and $-\frac{3}{\sqrt{x}} = -3x^{-1/2}$, applying the power rule and trigonometric rule gives the correct result.

> **Exam tip:** Differentiate your final antiderivative to catch sign errors before moving on in a problem.

## Common pitfalls

- **Wrong:** $\int x^{-1} dx = \frac{x^{0}}{0} + C$, leaving it undefined or incorrectly writing it as $1 + C$.
  - Why it fails: Students automatically apply the power rule without remembering the $n \neq -1$ exception.
  - Correct: Always check if $n = -1$ before applying the power rule, and use $\int \frac{1}{x} dx = \ln|x| + C$ for this case.
- **Wrong:** $\int 3^x dx = \frac{3^{x+1}}{x+1} + C$, treating the base 3 as a power of $x$ and misapplying the power rule.
  - Why it fails: Confusion between exponential functions (variable exponent, constant base) and power functions (constant exponent, variable base).
  - Correct: Explicitly label functions as power ($x^n$) vs exponential ($a^x$) before integrating, and use the exponential antiderivative rule $\int a^x dx = \frac{a^x}{\ln a} + C$ for $a^x$.
- **Wrong:** Omitting $+C$ when asked for the general antiderivative, for example writing $\int 2 \cos x dx = 2 \sin x$.
  - Why it fails: Students forget that indefinite integrals require the arbitrary constant, especially when carrying the result into a later part of a problem.
  - Correct: Add $+C$ to every indefinite integral result immediately after integrating, before moving to the next step of the problem.
- **Wrong:** $\int \sin x dx = \cos x + C$, missing the negative sign.
  - Why it fails: Students memorize only the derivative rule and forget to reverse the sign correctly when integrating.
  - Correct: After integrating any trigonometric function, differentiate your result to confirm it matches the original integrand; for example, $\frac{d}{dx}[\cos x] = -\sin x \neq \sin x$, so you know a sign is wrong.
- **Wrong:** $\int \frac{1}{x^3} dx = \frac{x^{-2}}{-2} + C$, then incorrectly rewriting this as $-2x^{-2} + C = -\frac{2}{x^2} + C$.
  - Why it fails: Students make an algebra mistake when simplifying the fraction after applying the power rule.
  - Correct: After calculating $\frac{x^{n+1}}{n+1}$, double-check the arithmetic with the denominator: for $n=-3$, $n+1=-2$, so $\frac{x^{-2}}{-2} = -\frac{1}{2x^2}$, not $-\frac{2}{x^2}$.
- **Wrong:** $\int \frac{1}{x} dx = \ln x + C$, omitting the absolute value around $x$.
  - Why it fails: Students only work with positive $x$ in examples and forget that $\frac{1}{x}$ is defined for negative $x$, and $\ln x$ is not.
  - Correct: Always write the absolute value in the logarithm whenever you integrate $\frac{1}{x}$; it is required for full credit on the AP exam.

## Cheatsheet

| Category | Formula | Notes |
| --- | --- | --- |
| Power Rule | $\int x^n dx = \frac{x^{n+1}}{n+1} + C$ | Only for $n \neq -1$ |
| Reciprocal Rule | $\int \frac{1}{x} dx = \ln\|x\| + C$ | Absolute value required |
| Constant Multiple | $\int k f(x) dx = k \int f(x) dx$ | Any constant can be pulled out |
| Sum/Difference | $\int (f(x) \pm g(x)) dx = \int f(x) dx \pm \int g(x) dx$ | Integrate term-by-term |
| Natural Exponential | $\int e^x dx = e^x + C$ | Valid for all real $x$ |
| General Exponential | $\int a^x dx = \frac{a^x}{\ln a} + C$ | For $a>0, a \neq 1$ |
| Natural Logarithm | $\int \ln x dx = x \ln x - x + C$ | For $x>0$ |
| Sine | $\int \sin x dx = -\cos x + C$ | Remember the negative sign |
| Cosine | $\int \cos x dx = \sin x + C$ | No negative sign here |
| Secant Squared | $\int \sec^2 x dx = \tan x + C$ | Common in all integration problems |

## What's next

Mastery of these basic antiderivative rules is a non-negotiable foundation for every integration technique you will learn next in AP Calculus BC. All advanced integration methods (including u-substitution, integration by parts, and partial fractions) ultimately rely on rewriting a complex integrand to fit one of these basic rules. Without automatic recall of these rules, you will not be able to recognize the simplified form after applying advanced techniques, making all subsequent integration work much harder. Beyond integration techniques, these rules are required to solve separable differential equations, calculate accumulation functions, areas, volumes, and arc lengths—all of which are heavily tested on the AP exam.

- [Integration by parts](https://www.owlsprep.com/study/ap-calculus-bc-u6-integration-by-parts/)
- [Integration by substitution (u-substitution)](https://www.owlsprep.com/study/ap-calculus-bc-u6-integration-by-substitution/)
- [Integration with long division and completing the square](https://www.owlsprep.com/study/ap-calculus-bc-u6-integration-with-long-division-and/)

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