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Computing Indefinite Integrals

Your Antiderivative Toolkit

Now that you understand the notation, it's time to build fluency. Computing indefinite integrals means recognizing patterns and applying formulas from your "table of integrals."

The good news: you already know these formulas—they're just your derivative formulas read backwards!

The key skill: combining basic formulas using linearity (sum rule and constant multiple rule) to handle more complex integrands.

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Prerequisite Skills

graph LR
    subgraph "Prerequisites"
        A["Indefinite Integral<br/>Notation"]
        B["Power Rule<br/>Antiderivatives"]
        C["Trig<br/>Antiderivatives"]
    end

    subgraph "This Skill"
        D["Computing<br/>Indefinite Integrals"]
    end

    subgraph "Unlocks"
        E["Net Change<br/>Theorem"]
        F["u-Substitution"]
        G["Evaluating<br/>Definite Integrals"]
    end

    A --> D
    B --> D
    C --> D
    D --> E
    D --> F
    D --> G

    click A "indefinite-integral-notation.html"
    click D "computing-indefinite-integrals.html"
    click E "net-change-theorem.html"

    click B "../ch2-sec3/power-rule.html"
    click F "../ch4-sec5/u-substitution-definite.html"
    click G "../ch4-sec2/definite-integral-evaluation.html"

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Before You Start

Prerequisite Check — Can you answer these?

From Indefinite Integral Notation:

1. What does $\int f(x)\, dx = F(x) + C$ mean in terms of derivatives?

From Power Rule Antiderivatives:

1. Evaluate $\int x^4\, dx$ (include $+C$).

1. Evaluate $\int \frac{1}{x^3}\, dx$ by first rewriting as a power.

From Trig Antiderivatives:

1. What is $\int \cos x\, dx$?

1. What is $\int \sec^2 x\, dx$?

Check Your Answers

1. It means $F'(x) = f(x)$ — the derivative of $F$ is $f$.

1. $\int x^4\, dx = \frac{x^5}{5} + C$

1. $\int x^{-3}\, dx = \frac{x^{-2}}{-2} + C = -\frac{1}{2x^2} + C$

1. $\int \cos x\, dx = \sin x + C$

1. $\int \sec^2 x\, dx = \tan x + C$

If these feel unfamiliar, review the prerequisite pages before continuing.

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Quick Reference

Property	Value
Concept	Indefinite Integrals & Net Change
Chapter	Chapter 4, Section 4
Difficulty	Intermediate
Time	~20 minutes

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Key Concepts

The Table of Indefinite Integrals

Power Functions

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

$$\int k\, dx = kx + C \quad \text{(constant)}$$

Trigonometric Functions

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

Linearity Rules (The Power Tools)

These let you break complex integrals into simpler pieces:

Constant Multiple Rule: $$\int c \cdot f(x)\, dx = c \int f(x)\, dx$$

Sum/Difference Rule: $$\int [f(x) \pm g(x)]\, dx = \int f(x)\, dx \pm \int g(x)\, dx$$

Strategy: Always simplify the integrand FIRST, then apply linearity to separate terms, then integrate term by term.

The "Simplify First" Principle

Many integrands look complicated but become easy after algebraic simplification:

Example: $\int \frac{x^3 + 2x}{x}\, dx$

Don't try to integrate this directly! Simplify first: $$= \int \left(x^2 + 2\right)\, dx = \frac{x^3}{3} + 2x + C$$

Common Patterns to Recognize

Pattern	Strategy
$\frac{a + b}{c}$	Split into $\frac{a}{c} + \frac{b}{c}$
$x^a \cdot x^b$	Combine to $x^{a+b}$
$\sqrt{x} = x^{1/2}$	Rewrite as power
$\frac{1}{x^n} = x^{-n}$	Rewrite as power
$\sqrt[n]{x^m} = x^{m/n}$	Rewrite as power

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Practice Problems

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Mastery Checklist

• [ ] I know the power rule for integration: $\int x^n\, dx = \frac{x^{n+1}}{n+1} + C$
• [ ] I know the antiderivatives of the six basic trig functions
• [ ] I can use linearity to break up sums and pull out constants
• [ ] I always simplify the integrand before integrating when possible
• [ ] I can convert roots and fractions to powers before integrating

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Mental Model

Integration as "Reverse Engineering"

Imagine each integral formula as a puzzle piece. The question is: "What function, when differentiated, gives me this?"

Your table of integrals is your collection of solved puzzles. For complex problems:

1. Simplify — reduce to basic puzzle pieces
2. Separate — use linearity to work on one piece at a time
3. Match — find each piece in your table
4. Assemble — combine answers with a single $+C$

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Common Errors to Avoid

Error	Correction
$\int 3x^2\, dx = 3x^3 + C$	Should be $x^3 + C$ (the 3 divides out)
$\int x^{-1}\, dx = \frac{x^0}{0}$	Power rule doesn't work for $n = -1$! Use $\ln\vert x\vert + C$
Multiple $+C$'s	Only ONE $+C$ per integral
Forgetting to simplify	$\int \frac{x^2 + 1}{x}\, dx \neq$ hard! Simplify first.

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Connections

Looking back:

• Indefinite Integral Notation — what the symbol means
• Antiderivatives — where these formulas come from

Looking ahead:

• Net Change Theorem — interpreting definite integrals
• u-Substitution — for when algebra alone isn't enough

Real-world connections:

• Computing indefinite integrals is the foundation for solving differential equations in physics and engineering
• Every definite integral computation starts by finding an indefinite integral

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Previous	Up	Next
Indefinite Integral Notation	Skills Index	Net Change Theorem

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Last updated: 2026-01-22