# Antiderivatives and indefinite integrals (basic rules)

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

This sub-topic introduces core definitions of antiderivatives and indefinite integrals, plus basic integration rules for algebraic, exponential, and trigonometric functions. It is the foundation for all further integration topics in AP Calculus AB.

**Prerequisites:** Derivative rules for polynomials, exponentials, and trigonometric functions; Basic algebraic manipulation of polynomial and rational terms; Chain rule for derivatives

## Learning objectives

- Define antiderivatives and indefinite integrals
- Apply basic integration rules to algebraic functions
- Find particular antiderivatives using initial conditions
- Integrate basic exponential and trigonometric functions

## Antiderivatives, Indefinite Integrals, and the Constant of Integration

**Antiderivative** — A differentiable function $F(x)$ is an antiderivative of $f(x)$ on an interval if $F'(x) = f(x)$ for all $x$ in the interval.

**Indefinite Integral** — The general representation of all possible antiderivatives of $f(x)$, including an arbitrary constant of integration $+C$. Outputs a family of functions, not a numerical value.

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

Because the derivative of any constant is zero, if $F(x)$ is an antiderivative of $f(x)$, then $F(x) + C$ for any real constant $C$ is also an antiderivative. The differential $dx$ in the notation explicitly identifies the variable of integration. If given an initial condition (a point the antiderivative must pass through), you can solve for a specific value of $C$ to get a particular antiderivative.

**Worked example:** Given $f(x) = 6x^2$, (1) write the general indefinite integral of $f(x)$, and (2) find the particular antiderivative that satisfies $F(1) = 10$.

1. Recall that the derivative of $x^3$ is $3x^2$, so the antiderivative of $x^2$ is $\frac{x^3}{3} + C$.
2. Apply the constant multiple rule to get the general indefinite integral:

   $$\int 6x^2 \, dx = 6 \cdot \frac{x^3}{3} + C = 2x^3 + C$$
3. Substitute the initial condition $F(1) = 10$ into the general form:

   $$F(1) = 2(1)^3 + C = 2 + C = 10$$
4. Solve for $C$: $C = 10 - 2 = 8$.
5. The particular antiderivative is:

   $$F(x) = 2x^3 + 8$$

> **Exam tip:** Always check your antiderivative by differentiating it! If the derivative of your result equals the original integrand, you know your work is correct. This catches 90% of common sign and arithmetic errors on the exam.

## Algebraic Basic Integration Rules

Algebraic integration rules directly mirror differentiation rules, reversed for integration. The constant multiple rule states $\int k \cdot f(x) \, dx = k \int f(x) \, dx$ for any constant $k$, and the sum/difference rule states $\int [f(x) \pm g(x)] dx = \int f(x) dx \pm \int g(x) dx$. The most widely used rule is the power rule for integration.

**Power Rule for Integration** — Reverses the power rule for derivatives. The key exception is when $n=-1$, which uses a logarithmic antiderivative rule. Always rewrite roots and reciprocals as $x^n$ form before applying the rule: $\frac{1}{x^3} = x^{-3}$ and $\sqrt{x} = x^{1/2}$.

*Notation:* \int x^n dx = \frac{x^{n+1}}{n+1} + C \quad n \neq -1

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

1. Rewrite all terms to match the $x^n$ form:

   $$\frac{2}{\sqrt{x}} = 2x^{-1/2}, \quad 5 = 5x^0 \implies \int \left( 4x^3 - 2x^{-1/2} + 5x^0 \right) dx$$
2. Split the integral using sum/difference and constant multiple rules:

   $$4 \int x^3 dx - 2 \int x^{-1/2} dx + 5 \int x^0 dx$$
3. Apply the power rule to each term:

   $$4 \cdot \frac{x^{4}}{4} - 2 \cdot \frac{x^{1/2}}{1/2} + 5 \cdot \frac{x^1}{1} + C$$
4. Simplify the final result:

   $$x^4 - 4\sqrt{x} + 5x + C$$
5. Verify by differentiation: the derivative of the result matches the original integrand.

> **Exam tip:** Always rewrite roots and reciprocals as power terms before applying the power rule. Skipping this step is the most common cause of miscalculating the exponent in the power rule on AP exams.

## Basic Transcendental Antiderivatives (Exponential and Trigonometric)

All rules for non-algebraic functions are direct reverses of derivative rules you already know. The key results for AP Calculus AB are:

- Exponentials: $\int e^x dx = e^x + C$; $\int a^x dx = \frac{a^x}{\ln a} + C$ for $a>0, a \neq 1$
- Reciprocal ($n=-1$ exception): $\int \frac{1}{x} dx = \ln|x| + C$. The absolute value covers all non-zero $x$.
- Trigonometric: $\int \cos x dx = \sin x + C$, $\int \sin x dx = -\cos x + C$, $\int \sec^2 x dx = \tan x + C$, $\int \sec x \tan x dx = \sec x + C$

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

1. Split the integral into separate terms using the sum/difference rule:

   $$3 \int e^x dx + 2 \int \sin x dx - \int \sec x \tan x dx + 4 \int \frac{1}{x} dx$$
2. Apply the corresponding antiderivative rule to each term:

   $$3(e^x) + 2(-\cos x) - (\sec x) + 4(\ln|x|) + C$$
3. Simplify to get the final result:

   $$3e^x - 2\cos x - \sec x + 4\ln|x| + C$$
4. Verify by differentiation to confirm the result matches the original integrand.

> **Exam tip:** If you are unsure about the sign of a trigonometric antiderivative, take 10 seconds to differentiate your result to confirm it matches the original integrand. FRQ graders deduct points for incorrect signs, so this check is well worth the time.

## AP-Style Concept Check Practice

**Check your understanding**

Test your understanding with these AP-style problems:

1. Which of the following is the general indefinite integral $\int \left( \frac{3x^2 - 2\sqrt{x}}{x} \right) dx$?

   - $\frac{3}{2}x^2 - 4\sqrt{x} + C$
   - $3x^2 - 2\sqrt{x} + C$
   - $3x - 4\sqrt{x} + C$
   - $x^3 - 2x^2 + C$

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

   *Why:* Correct: Simplify the integrand to $3x - 2x^{-1/2}$ then apply the power rule to get this result.

**Worked example:** Let $f''(x) = 2\cos x + 12x^2$. It is known that $f'(0) = 2$ and $f(0) = 5$. (a) Find the general $f'(x)$, (b) find particular $f'(x)$, (c) find particular $f(x)$.

1. Integrate $f''(x)$ for general $f'(x)$:

   $$f'(x) = \int (2\cos x + 12x^2) dx = 2\sin x + 4x^3 + C$$
2. Substitute $f'(0) = 2$ to solve for $C$:

   $$f'(0) = 0 + 0 + C = 2 \implies C = 2 \implies f'(x) = 2\sin x + 4x^3 + 2$$
3. Integrate $f'(x)$ for general $f(x)$:

   $$f(x) = \int (2\sin x + 4x^3 + 2) dx = -2\cos x + x^4 + 2x + K$$
4. Substitute $f(0) = 5$ to solve for $K$:

   $$f(0) = -2(1) + K = 5 \implies K=7 \implies f(x) = x^4 - 2\cos x + 2x + 7$$

**Worked example:** Velocity of a projectile is $v(t) = 3t^2 - 2t + 4$ m/s. Initial position at $t=0$ is 10 m. Find position function $s(t)$ and position at $t=3$ s.

1. Position is the antiderivative of velocity:

   $$s(t) = \int (3t^2 - 2t + 4) dt$$
2. Apply power rule to get general form:

   $$s(t) = t^3 - t^2 + 4t + C$$
3. Use initial condition $s(0)=10$ to get $C=10$, so:

   $$s(t) = t^3 - t^2 + 4t + 10$$
4. Evaluate at $t=3$:

   $$s(3) = 27 - 9 + 12 + 10 = 40$$
5. The projectile is 40 meters from the origin at $t=3$ seconds.

## Common pitfalls

- **Wrong:** Writing $\int x^{-1} dx = \frac{x^{0}}{0} + C$, leaving the answer undefined or incorrectly simplifying $\frac{1}{0}$ to 1.
  - Why it fails: Students blindly apply the power rule without remembering the exception for $n=-1$.
  - Correct: Always check if $n=-1$ before applying the power rule; if yes, use $\int \frac{1}{x} dx = \ln|x| + C$ instead.
- **Wrong:** Substituting the initial condition into the original integrand $f(x)$ instead of the general antiderivative to solve for $C$. For example, given $f(x)=6x^2$ and $F(1)=10$, calculating $6(1)^2=10+C$ to get $C=-4$.
  - Why it fails: Students confuse the original function with its antiderivative.
  - Correct: Always find the general antiderivative first, then substitute the given $x$-value into the general antiderivative to solve for $C$.
- **Wrong:** Omitting the absolute value in $\ln|x|+C$ and writing $\ln x + C$ instead.
  - Why it fails: Students memorize the rule without remembering the domain of $\frac{1}{x}$ includes negative $x$.
  - Correct: Always write the absolute value around the logarithm argument for $\int \frac{1}{x} dx$, even if the problem doesn't specify the domain, to get full credit on FRQs.
- **Wrong:** Writing $\int (2x+1)^2 dx = \frac{(2x+1)^3}{3} + C$, without accounting for the inner coefficient of $x$.
  - Why it fails: Students apply the basic power rule directly to composite functions before learning u-substitution.
  - Correct: Expand polynomial powers first when using only basic integration rules, or wait for u-substitution to integrate non-basic composite functions.
- **Wrong:** Adding a separate constant of integration to every term, e.g., $\int 4x^3 - 2x dx = x^4 + C_1 - x^2 + C_2$.
  - Why it fails: Students think each term needs its own constant.
  - Correct: Add only one constant of integration $+C$ at the end of the entire antiderivative, since the sum of multiple arbitrary constants is just one arbitrary constant.
- **Wrong:** Writing $\int 2^x dx = \frac{2^x}{2} + C$.
  - Why it fails: Students mix up the power rule for polynomials with the exponential integration rule.
  - Correct: Remember $\int a^x dx = \frac{a^x}{\ln a} + C$; the denominator is the natural log of the base, not the base itself.

## Cheatsheet

| Category | Formula | Notes |
| --- | --- | --- |
| General Indefinite Integral | $\int f(x) dx = F(x) + C$, where $F'(x)=f(x)$ | One $+C$ per full integral; $C$ is arbitrary constant |
| Power Rule | $\int x^n dx = \frac{x^{n+1}}{n+1} + C$ | Only applies for $n \neq -1$; rewrite roots/reciprocals as powers first |
| Reciprocal Rule | $\int \frac{1}{x} dx = \ln\|x\| + C$ | Keep absolute value to cover negative $x$ |
| Constant Multiple Rule | $\int k f(x) dx = k \int f(x) dx$ | Works for any constant real $k$ |
| Sum/Difference Rule | $\int [f(x) \pm g(x)] dx = \int f(x) dx \pm \int g(x) dx$ | Split integrals into individual terms to apply basic rules |
| Exponential Rules | $\int e^x dx = e^x + C$; $\int a^x dx = \frac{a^x}{\ln a} + C$ | $a>0, a \neq 1$; do not confuse with polynomial power rule |
| Basic Trig Rules 1 | $\int \cos x dx = \sin x + C$; $\int \sin x dx = -\cos x + C$ | Remember negative sign for sine's antiderivative |
| Basic Trig Rules 2 | $\int \sec^2 x dx = \tan x + C$; $\int \sec x \tan x dx = \sec x + C$ | All rules are reversed derivative rules |

## What's next

Mastering the basic antiderivative rules in this topic is a non-negotiable prerequisite for every integration topic that follows in AP Calculus AB. Next, you will learn u-substitution, the core technique for integrating composite functions, which relies entirely on your ability to quickly recall and apply these basic antiderivative rules after completing the substitution step. Without automatic mastery of these rules, you will not be able to focus on the substitution logic, and will make frequent calculation errors on even routine problems. This topic is also the foundation for solving separable differential equations, calculating accumulated change, finding areas between curves, and solving kinematics motion problems—all high-weight content on the AP exam.

- [Integration by substitution (u-sub)](https://www.owlsprep.com/study/ap-calculus-ab-u6-integration-by-substitution/)
- [Integration with long division and completing the square](https://www.owlsprep.com/study/ap-calculus-ab-u6-integration-with-long-division-and/)
- [Selecting techniques for antidifferentiation](https://www.owlsprep.com/study/ap-calculus-ab-u6-selecting-techniques-for-antidifferentiation/)

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