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.
Prerequisite Skills
Before You Start
Prerequisite Check: Can you answer these?
From Indefinite Integral Notation:
- What does $\int f(x)\, dx = F(x) + C$ mean in terms of derivatives?
From Power Rule Antiderivatives:
Evaluate $\int x^4\, dx$ (include $+C$).
Evaluate $\int \frac{1}{x^3}\, dx$ by first rewriting as a power.
From Trig Antiderivatives:
What is $\int \cos x\, dx$?
What is $\int \sec^2 x\, dx$?
Check Your Answers
It means $F'(x) = f(x)$: the derivative of $F$ is $f$.
$\int x^4\, dx = \frac{x^5}{5} + C$
$\int x^{-3}\, dx = \frac{x^{-2}}{-2} + C = -\frac{1}{2x^2} + C$
$\int \cos x\, dx = \sin x + C$
$\int \sec^2 x\, dx = \tan x + C$
If these feel unfamiliar, review the prerequisite pages before continuing.
Quick Reference
| Property | Value |
|---|---|
| Concept | Indefinite Integrals & Net Change |
| Chapter | Chapter 4, Section 4 |
| Difficulty | Intermediate |
| Time | ~20 minutes |
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
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 |
Practice Problems
Find the general indefinite integral:
$$\int (3x^2 + 4x + 1)\, dx$$
Find the general indefinite integral:
$$\int (4\sec^2 x - 2\sin x)\, dx$$
Find the general indefinite integral:
$$\int \frac{2t^3 + t^2\sqrt{t}}{t^2}\, dt$$
Find the general indefinite integral:
$$\int \frac{\cos\theta}{\sin^2\theta}\, d\theta$$
Hint: Rewrite using trig identities.
The power rule states $\int x^n\, dx = \frac{x^{n+1}}{n+1} + C$ for $n \neq -1$.
(a) What happens algebraically if you try to apply this formula when $n = -1$?
(b) We know $\frac{d}{dx}[\ln\vert x\vert ] = \frac{1}{x}$. Use this to explain what $\int x^{-1}\, dx$ actually equals.
(c) Deeper question: The function $f(x) = \frac{x^{n+1}}{n+1}$ is continuous for all $n > -1$ and all $n < -1$. What happens to this function as $n \to -1$? Does it “approach” $\ln\vert x\vert $ in any sense?
Common Misconceptions
the integral of a product equals the product of the individual integrals.
This is the multiplicative-not-additive error. The linearity rules state $\int(f + g) = \int f + \int g$ and $\int cf = c\int f$, but there is no product rule for integrals: $\int f(x)g(x)\,dx \neq \left(\int f(x)\,dx\right)\left(\int g(x)\,dx\right)$ in general. For instance, $\int x\cos x\,dx$ is not $\left(\int x\,dx\right)\left(\int \cos x\,dx\right) = \frac{x^2}{2}\sin x + C$. The correct method (integration by parts) gives $x\sin x + \cos x + C$, a completely different function.
Mastery Checklist
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:
- Simplify: reduce to basic puzzle pieces
- Separate: use linearity to work on one piece at a time
- Match: find each piece in your table
- Assemble: combine answers with a single $+C$
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. |
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
| Previous | Up | Next |
|---|---|---|
| Indefinite Integral Notation | Skills Index | Net Change Theorem |
Last updated: 2026-01-22