Unit 6.8 – Finding Antiderivatives and Indefinite Integrals: Basic Rules and Notation

AP® Calculus AB & BC | Reversing Differentiation

Why This Matters: Antiderivatives (also called indefinite integrals) are the reverse process of differentiation! If derivatives measure rates of change, antiderivatives find the original function from its rate of change. This is fundamental to solving differential equations, computing areas, finding displacement from velocity, and countless other applications. Mastering the basic integration rules is essential—they're the building blocks for all integration techniques you'll learn. The constant of integration (+C) is critical and reflects the fact that many functions have the same derivative!

📘 Definitions and Notation

ANTIDERIVATIVE DEFINITION

A function \(F(x)\) is called an antiderivative of \(f(x)\) if:

\[ F'(x) = f(x) \]

In words: \(F\) is an antiderivative of \(f\) if the derivative of \(F\) equals \(f\).

INDEFINITE INTEGRAL NOTATION

The indefinite integral of \(f(x)\) is written as:

\[ \int f(x) \, dx = F(x) + C \]

Components of Indefinite Integral:

\(\int\) = integral sign
\(f(x)\) = integrand (function being integrated)
\(dx\) = indicates the variable of integration
\(F(x)\) = antiderivative of \(f(x)\)
\(C\) = constant of integration (ALWAYS include!)

🔑 The Constant of Integration (+C):

If \(F(x)\) is one antiderivative of \(f(x)\), then all antiderivatives are of the form \(F(x) + C\) where \(C\) is any constant.

Why? Because the derivative of a constant is zero: \(\frac{d}{dx}[F(x) + C] = F'(x) + 0 = f(x)\)

⚠️ NEVER forget the +C in indefinite integrals!

📐 Basic Integration Rules

Essential Antiderivative Formulas

Core Integration Formulas
Function \(f(x)\)	Antiderivative \(\int f(x)\,dx\)	Notes
\(k\) (constant)	\(kx + C\)	Constant rule
\(x^n\) \((n \neq -1)\)	\(\frac{x^{n+1}}{n+1} + C\)	Power rule (increase exponent by 1, divide)
\(\frac{1}{x}\) or \(x^{-1}\)	\(\ln\|x\| + C\)	Special case when \(n = -1\)
\(e^x\)	\(e^x + C\)	Exponential stays the same!
\(a^x\)	\(\frac{a^x}{\ln a} + C\)	General exponential \((a > 0, a \neq 1)\)
\(\sin x\)	\(-\cos x + C\)	Note the negative!
\(\cos x\)	\(\sin x + C\)	Positive (no negative)
\(\sec^2 x\)	\(\tan x + C\)	Derivative of tangent
\(\csc^2 x\)	\(-\cot x + C\)	Note the negative!
\(\sec x \tan x\)	\(\sec x + C\)	Derivative of secant
\(\csc x \cot x\)	\(-\csc x + C\)	Note the negative!

💪 The Power Rule for Integration

Power Rule for Integration:

\[ \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \quad \text{(for } n \neq -1\text{)} \]

Steps:

Add 1 to the exponent: \(n \to n+1\)
Divide by the new exponent: \(\frac{x^{n+1}}{n+1}\)
Add the constant of integration: \(+ C\)

Examples of Power Rule:

Example 1: \(\int x^3 \, dx\)

\[ = \frac{x^{3+1}}{3+1} + C = \frac{x^4}{4} + C \]

Example 2: \(\int x \, dx\) (same as \(\int x^1 \, dx\))

\[ = \frac{x^{1+1}}{1+1} + C = \frac{x^2}{2} + C \]

Example 3: \(\int \frac{1}{x^2} \, dx\) (rewrite as \(\int x^{-2} \, dx\))

\[ = \frac{x^{-2+1}}{-2+1} + C = \frac{x^{-1}}{-1} + C = -\frac{1}{x} + C \]

Example 4: \(\int \sqrt{x} \, dx\) (rewrite as \(\int x^{1/2} \, dx\))

\[ = \frac{x^{1/2+1}}{1/2+1} + C = \frac{x^{3/2}}{3/2} + C = \frac{2x^{3/2}}{3} + C \]

🔧 Properties of Indefinite Integrals

Linearity Properties

Property 1: Constant Multiple Rule

\[ \int k \cdot f(x) \, dx = k \int f(x) \, dx \]

Constants factor out of integrals

Property 2: Sum/Difference Rule

\[ \int [f(x) \pm g(x)] \, dx = \int f(x) \, dx \pm \int g(x) \, dx \]

Integral of sum = sum of integrals

Combined (Linearity):

\[ \int [af(x) + bg(x)] \, dx = a\int f(x)\,dx + b\int g(x)\,dx \]

📖 Comprehensive Worked Examples

Example 1: Basic Polynomial

Problem: Find \(\int (3x^2 + 5x - 7) \, dx\)

Solution:

Step 1: Split using sum/difference rule

\[ \int (3x^2 + 5x - 7) \, dx = \int 3x^2\,dx + \int 5x\,dx - \int 7\,dx \]

Step 2: Factor out constants

\[ = 3\int x^2\,dx + 5\int x\,dx - 7\int 1\,dx \]

Step 3: Apply power rule

\[ = 3 \cdot \frac{x^3}{3} + 5 \cdot \frac{x^2}{2} - 7x + C \]

Step 4: Simplify

\[ = x^3 + \frac{5x^2}{2} - 7x + C \]

Answer: \(x^3 + \frac{5x^2}{2} - 7x + C\)

Example 2: Fractional and Negative Exponents

Problem: Find \(\int \left(\frac{2}{x^3} + \sqrt[3]{x}\right) \, dx\)

Solution:

Step 1: Rewrite using exponents

\[ \int \left(2x^{-3} + x^{1/3}\right) \, dx \]

Step 2: Apply rules

\[ = 2\int x^{-3}\,dx + \int x^{1/3}\,dx \]

Step 3: Power rule

\[ = 2 \cdot \frac{x^{-2}}{-2} + \frac{x^{4/3}}{4/3} + C \]

\[ = -x^{-2} + \frac{3x^{4/3}}{4} + C \]

Step 4: Rewrite with positive exponents

\[ = -\frac{1}{x^2} + \frac{3x^{4/3}}{4} + C \]

Answer: \(-\frac{1}{x^2} + \frac{3x^{4/3}}{4} + C\) or \(-\frac{1}{x^2} + \frac{3\sqrt[3]{x^4}}{4} + C\)

Example 3: Trigonometric Functions

Problem: Find \(\int (4\sin x - 3\cos x + \sec^2 x) \, dx\)

Solution:

Apply linearity and use trig formulas:

\[ = 4\int \sin x\,dx - 3\int \cos x\,dx + \int \sec^2 x\,dx \]

\[ = 4(-\cos x) - 3(\sin x) + \tan x + C \]

\[ = -4\cos x - 3\sin x + \tan x + C \]

Answer: \(-4\cos x - 3\sin x + \tan x + C\)

Example 4: Mixed Functions

Problem: Find \(\int \left(e^x + \frac{5}{x} - 2^x\right) \, dx\)

Solution:

Split and apply formulas:

\[ = \int e^x\,dx + 5\int \frac{1}{x}\,dx - \int 2^x\,dx \]

\[ = e^x + 5\ln|x| - \frac{2^x}{\ln 2} + C \]

Answer: \(e^x + 5\ln|x| - \frac{2^x}{\ln 2} + C\)

✅ Verifying Your Answer

How to Check Your Antiderivative:

Take the derivative of your answer! If you get back the original function, you're correct.

Example: Verify that \(\int 3x^2\,dx = x^3 + C\)

Check: \(\frac{d}{dx}[x^3 + C] = 3x^2 + 0 = 3x^2\) ✓

💡 Essential Tips & Tricks

✅ Integration Success Tips:

Always add +C: For indefinite integrals (no limits)
Rewrite before integrating: Convert roots to fractional exponents, fractions to negative exponents
Factor out constants: Makes integration much easier
Check your work: Differentiate your answer to verify
Watch for \(n = -1\): Power rule doesn't work; use \(\ln|x|\) instead
Memorize trig antiderivatives: Especially the negative signs!
Don't forget absolute value: \(\int \frac{1}{x}\,dx = \ln|x| + C\)

📝 Common Rewrites:

\(\sqrt{x} = x^{1/2}\)
\(\sqrt[n]{x} = x^{1/n}\)
\(\frac{1}{x^n} = x^{-n}\)
\(\frac{1}{\sqrt{x}} = x^{-1/2}\)
\(\sqrt{x^3} = x^{3/2}\)

❌ Common Mistakes to Avoid

Mistake 1: Forgetting the +C in indefinite integrals
Mistake 2: Using power rule for \(n = -1\) (should be \(\ln|x|\))
Mistake 3: Forgetting negative signs in trig antiderivatives (\(\sin x\), \(\cot x\), \(\csc x\))
Mistake 4: Not factoring out constants before integrating
Mistake 5: Forgetting to divide by new exponent in power rule
Mistake 6: Thinking \(\int \frac{1}{x}\,dx = \frac{1}{0} = \text{undefined}\) (it's \(\ln|x| + C\)!)
Mistake 7: Writing \(\ln x\) instead of \(\ln|x|\)
Mistake 8: Not simplifying fractions in exponents
Mistake 9: Trying to integrate products/quotients like derivatives (no product/quotient rule for integrals!)
Mistake 10: Not rewriting radicals and fractions as powers first

📝 Practice Problems

Set A: Basic Integration

\(\int 5x^4 \, dx\)
\(\int (x^3 - 6x^2 + 4) \, dx\)
\(\int (3\sqrt{x} + \frac{2}{x^2}) \, dx\)

Answers:

\(x^5 + C\)
\(\frac{x^4}{4} - 2x^3 + 4x + C\)
\(2x^{3/2} - \frac{2}{x} + C\)

Set B: Mixed Functions

\(\int (2e^x - 3\sin x) \, dx\)
\(\int \left(\frac{4}{x} + \sec^2 x\right) \, dx\)
\(\int (x^{-3} + x^{1/3}) \, dx\)

Answers:

\(2e^x + 3\cos x + C\)
\(4\ln|x| + \tan x + C\)
\(-\frac{1}{2x^2} + \frac{3x^{4/3}}{4} + C\)

✏️ AP® Exam Success Tips

What AP® Graders Look For:

Always include +C: Points deducted if missing
Show work: Don't skip steps, even if easy
Simplify final answer: Combine like terms, write with positive exponents when possible
Check dimensional consistency: Units should make sense
Verify if time permits: Take derivative to check

⚡ Ultimate Quick Reference

ESSENTIAL FORMULAS - MEMORIZE THESE!

Quick Reference Table
\(f(x)\)	\(\int f(x)\,dx\)
\(x^n\) \((n \neq -1)\)	\(\frac{x^{n+1}}{n+1} + C\)
\(\frac{1}{x}\)	\(\ln\|x\| + C\)
\(e^x\)	\(e^x + C\)
\(\sin x\)	\(-\cos x + C\)
\(\cos x\)	\(\sin x + C\)
\(\sec^2 x\)	\(\tan x + C\)

Master the Basics! Antiderivatives (indefinite integrals) reverse differentiation. The power rule \(\int x^n\,dx = \frac{x^{n+1}}{n+1} + C\) is your workhorse (but not for \(n=-1\), use \(\ln|x|\)). Integration is linear: constants factor out, and you can split sums/differences. Always add +C for indefinite integrals—it represents the family of all antiderivatives. Key formulas to memorize: \(\int e^x\,dx = e^x + C\), \(\int \sin x\,dx = -\cos x + C\) (negative!), \(\int \cos x\,dx = \sin x + C\), \(\int \sec^2 x\,dx = \tan x + C\). Before integrating, rewrite roots as fractional exponents and fractions as negative exponents. Verify your answer by differentiating—you should get back the original function. These basic rules are the foundation for all integration techniques. Practice until they're automatic! 🎯✨