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Antiderivatives &
Indefinite Integrals

Master the basic rules and notation for reversing differentiation: find general antiderivatives using the Power Rule, constant multiples, and sums.

AP Calculus AB

What Is an Antiderivative?

Differentiation takes a function and produces its rate of change. An antiderivative reverses this process: given a function \( f(x) \), we look for a function \( F(x) \) whose derivative equals \( f(x) \).

For example, \( F(x) = x^3 \) is an antiderivative of \( f(x) = 3x^2 \) because \( \frac{d}{dx}[x^3] = 3x^2 \).

The Family of Antiderivatives

An antiderivative is never unique. If \( F(x) \) is an antiderivative of \( f(x) \), then so is \( F(x) + C \) for any constant \( C \), because the derivative of a constant is zero.

x y \(F(x) + C_1\) \(F(x)\) \(F(x) + C_2\)

Indefinite Integral Notation

We write the most general antiderivative of \( f(x) \) using the indefinite integral symbol:

\( \int \)
Integral sign: indicates antidifferentiation
\( f(x) \)
Integrand: the function to antidifferentiate
\( dx \)
Differential: tells us the variable of integration
\( + C \)
Constant of integration: accounts for all antiderivatives

Why Antiderivatives Matter

Antiderivatives are the foundation of the Fundamental Theorem of Calculus. They let us compute areas, solve differential equations, and recover original quantities from their rates of change.

Area Under Curves
Definite integrals use antiderivatives to calculate exact areas
Motion Problems
Integrate acceleration to find velocity, velocity to find position
Differential Equations
Solving \( \frac{dy}{dx} = f(x) \) means finding an antiderivative

The Power Rule for Integration

This rule reverses the Power Rule for differentiation. Add 1 to the exponent, then divide by the new exponent. The restriction \( n \neq -1 \) exists because division by zero is undefined.

Essential Integration Rules

1
Constant Multiple: \( \displaystyle\int k \cdot f(x)\, dx = k \int f(x)\, dx \)
2
Sum / Difference: \( \displaystyle\int [f(x) \pm g(x)]\, dx = \int f(x)\, dx \pm \int g(x)\, dx \)
3
Constant Function: \( \displaystyle\int k\, dx = kx + C \)
4
Special case \( n = -1 \): \( \displaystyle\int x^{-1}\, dx = \int \frac{1}{x}\, dx = \ln|x| + C \)

Derivatives vs. Antiderivatives

Derivative Antiderivative
Operation \( \frac{d}{dx} \) \( \int \cdots dx \)
Power Rule Multiply, then subtract 1 Add 1, then divide
Constants Disappear: \( \frac{d}{dx}[C] = 0 \) Reappear: \( + C \)
Uniqueness Unique result Family of functions
Linearity Yes Yes

How to Find an Indefinite Integral

1
Rewrite the integrand so every term is in the form \( k \cdot x^n \). Convert roots, fractions, and reciprocals to power notation.
2
Split the integral using the Sum/Difference Rule: integrate each term separately.
3
Factor out any constant coefficients using the Constant Multiple Rule.
4
Apply the Power Rule to each \( x^n \) term: add 1 to the exponent, divide by the new exponent.
5
Add \( + C \) at the end. Only one constant of integration is needed for the entire expression.

Example 1: Basic Polynomial

Find \( \displaystyle\int (3x^2 + 5x - 4)\, dx \).

1
Split by term: \( \displaystyle\int 3x^2\, dx + \int 5x\, dx - \int 4\, dx \)
2
Factor constants: \( 3\!\displaystyle\int x^2\, dx + 5\!\int x^1\, dx - 4\!\int dx \)
3
Apply Power Rule: \( 3 \cdot \dfrac{x^3}{3} + 5 \cdot \dfrac{x^2}{2} - 4x \)
4
Simplify: \( x^3 + \dfrac{5x^2}{2} - 4x + C \)

Example 2: Rewriting First

Find \( \displaystyle\int \left(\sqrt{x} + \frac{3}{x^2}\right) dx \).

1
Rewrite in power form: \( \sqrt{x} = x^{1/2} \) and \( \frac{3}{x^2} = 3x^{-2} \)
2
Integrate term-by-term: \( \displaystyle\int x^{1/2}\, dx + 3\!\int x^{-2}\, dx \)
3
Power Rule: \( \dfrac{x^{3/2}}{3/2} + 3 \cdot \dfrac{x^{-1}}{-1} \)
4
Simplify: \( \dfrac{2}{3}\,x^{3/2} - \dfrac{3}{x} + C \)

Example 3: Visual Verification

Find \( \displaystyle\int 2x\, dx \) and verify graphically.

x y 1 2 -1 -2 1 2 3 \(f(x) = 2x\) \(F(x) = x^2\)

Basic Trigonometric Integrals

Since integration reverses differentiation, we can read the trig derivative table backwards:

Integral Result
\( \int \cos x\, dx \) \( \sin x + C \)
\( \int \sin x\, dx \) \( -\cos x + C \)
\( \int \sec^2 x\, dx \) \( \tan x + C \)
\( \int \csc^2 x\, dx \) \( -\cot x + C \)
\( \int \sec x \tan x\, dx \) \( \sec x + C \)
\( \int \csc x \cot x\, dx \) \( -\csc x + C \)

Exponential and Logarithmic Integrals

\( \int e^x\, dx \)
\( = e^x + C \)
The exponential function is its own antiderivative.
\( \int \frac{1}{x}\, dx \)
\( = \ln|x| + C \)
This fills the gap where the Power Rule fails (\( n = -1 \)).

Common Mistakes to Avoid

✕ Forgetting \( +C \)
Every indefinite integral must include the constant of integration. Without it, you only have one antiderivative, not the general solution.
✕ Wrong Power Rule Direction
Students subtract 1 instead of adding 1, or multiply instead of dividing. Remember: integration is the reverse of differentiation.
✕ Not Rewriting First
Trying to integrate \( \sqrt{x} \) or \( \frac{1}{x^3} \) without converting to \( x^{1/2} \) or \( x^{-3} \) first leads to errors.
✕ Using Power Rule When \( n = -1 \)
\( \int x^{-1}\, dx \neq \frac{x^0}{0} \). Division by zero is undefined. The answer is \( \ln|x| + C \).

Quick Reference Table

Function \( f(x) \) Antiderivative \( \int f(x)\, dx \)
\( x^n \;\; (n \neq -1) \) \( \frac{x^{n+1}}{n+1} + C \)
\( \frac{1}{x} \) \( \ln|x| + C \)
\( e^x \) \( e^x + C \)
\( \cos x \) \( \sin x + C \)
\( \sin x \) \( -\cos x + C \)
\( \sec^2 x \) \( \tan x + C \)
\( k \) (constant) \( kx + C \)

Question 1 of 5

Evaluate \( \displaystyle\int 6x^2\, dx \).

Question 2 of 5

Evaluate \( \displaystyle\int \left(4x^3 - 2x + 7\right) dx \).

Question 3 of 5

Evaluate \( \displaystyle\int \frac{5}{\sqrt{x}}\, dx \).

Question 4 of 5

Which of the following equals \( \displaystyle\int \sin x\, dx \)?

Question 5 of 5

Why must we write \( + C \) in every indefinite integral?