Antiderivatives &
Indefinite Integrals
Master the reverse of differentiation: learn the notation, basic rules, and techniques for finding antiderivatives of common functions.
AP Calculus BC
What Is an Antiderivative?
An antiderivative of a function \( f(x) \) is any function \( F(x) \) whose derivative equals \( f(x) \). In other words:
For example, \( F(x) = x^3 \) is an antiderivative of \( f(x) = 3x^2 \) because \( \frac{d}{dx}[x^3] = 3x^2 \).
Differentiation Reversed
The Constant of Integration
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.
Indefinite Integral Notation
The indefinite integral of \( f(x) \) is written using the integral sign:
\( \int \)
The integral sign: tells you to find the antiderivative.
\( f(x) \)
The integrand: the function you are integrating.
\( dx \)
The differential: indicates the variable of integration.
The Power Rule for Integration
Raise the exponent by one, then divide by the new exponent. This is the reverse of the power rule for derivatives.
Example
\( \int x^4\, dx = \frac{x^5}{5} + C \)
Special case
\( \int 1\, dx = x + C \) (since \( x^0 = 1 \))
Essential Integration Rules
1
Constant Multiple: \( \int k \cdot f(x)\, dx = k \int f(x)\, dx \)
2
Sum / Difference: \( \int [f(x) \pm g(x)]\, dx = \int f(x)\, dx \pm \int g(x)\, dx \)
3
Exponential: \( \int e^x\, dx = e^x + C \)
4
Reciprocal: \( \int \frac{1}{x}\, dx = \ln|x| + C \)
Trigonometric Antiderivatives
| Integral | Result |
|---|---|
| \( \int \cos x\, dx \) | \( \sin x + C \) |
| \( \int \sin x\, dx \) | \( -\cos x + C \) |
| \( \int \sec^2 x\, dx \) | \( \tan x + C \) |
| \( \int \csc^2 x\, dx \) | \( -\cot x + C \) |
| \( \int \sec x \tan x\, dx \) | \( \sec x + C \) |
| \( \int \csc x \cot x\, dx \) | \( -\csc x + C \) |
Definite vs. Indefinite Integrals
| Indefinite | Definite | |
|---|---|---|
| Notation | \( \int f(x)\, dx \) | \( \int_a^b f(x)\, dx \) |
| Result | A family of functions | A number |
| Includes \( +C \)? | Yes | No |
| Bounds? | No | Yes (\( a \) to \( b \)) |
| Interpretation | General antiderivative | Net area under curve |
How to Find an Indefinite Integral
1
Simplify the integrand if possible: expand products, split fractions, rewrite radicals as fractional exponents.
2
Identify the rule that matches each term: power rule, exponential, trig, or reciprocal.
3
Apply the rule term by term, using linearity (sum/difference and constant multiple rules).
4
Add \( +C \) at the end: the constant of integration.
5
Verify by differentiating your answer: you should recover the original integrand.
Example 1: Polynomial
1
Integrate term by term: \( \int 3x^2\, dx - \int 5x\, dx + \int 4\, dx \)
2
Apply the power rule to each term: \( 3 \cdot \frac{x^3}{3} - 5 \cdot \frac{x^2}{2} + 4x \)
3
Simplify: \( x^3 - \frac{5x^2}{2} + 4x + C \)
Example 2: Rewriting First
1
Rewrite: \( \frac{2}{x^3} = 2x^{-3} \)
2
Power rule: \( 2 \cdot \frac{x^{-3+1}}{-3+1} = 2 \cdot \frac{x^{-2}}{-2} = -x^{-2} \)
3
Rewrite back: \( -\frac{1}{x^2} + C \)
Example 3: Mixed Functions
1
Rewrite: \( 4\sqrt{x} = 4x^{1/2} \)
2
Integrate each: \( 4 \cdot \frac{x^{3/2}}{3/2} + 3\sin x - \ln|x| \)
3
Simplify: \( \frac{8}{3}x^{3/2} + 3\sin x - \ln|x| + C \)
Visualizing \( f(x) \) and \( F(x) \)
Common Mistakes to Avoid
✕ Forgetting \( +C \)
An indefinite integral always requires the constant of integration. Without it, your answer represents only one antiderivative, not the full family.
✕ Wrong exponent in Power Rule
Students write \( \frac{x^n}{n} \) instead of \( \frac{x^{n+1}}{n+1} \). Remember: add one first, then divide by the new exponent.
✕ Using Power Rule when \( n = -1 \)
\( \int x^{-1}\, dx \neq \frac{x^0}{0} \). The power rule fails here. The correct 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: \( \int 6x^5\, dx \)
By the power rule: \( \int 6x^5\, dx = 6 \cdot \frac{x^6}{6} + C = x^6 + C \). Choice (a) is the derivative, not the integral. Choice (d) forgets to divide by 6.
Question 2 of 5
Evaluate: \( \int \left( 2\cos x + \frac{3}{x} \right) dx \)
\( \int \cos x\, dx = \sin x \) and \( \int \frac{1}{x}\, dx = \ln|x| \). So the answer is \( 2\sin x + 3\ln|x| + C \). Choice (a) has the wrong sign on \( \sin x \).
Question 3 of 5
What is the antiderivative of \( f(x) = \frac{4}{\sqrt{x}} \)?
Rewrite: \( \frac{4}{\sqrt{x}} = 4x^{-1/2} \). Apply the power rule: \( 4 \cdot \frac{x^{1/2}}{1/2} = 4 \cdot 2x^{1/2} = 8\sqrt{x} + C \). Choice (b) simplifies to \( 2\sqrt{x} \), which is incorrect.
Question 4 of 5
Why do we write \( +C \) in an indefinite integral?
Since \( \frac{d}{dx}[C] = 0 \), differentiating \( F(x) + C \) gives the same result as differentiating \( F(x) \). Thus infinitely many functions share the same derivative, and \( +C \) accounts for all of them.
Question 5 of 5
Evaluate: \( \int \left( e^x - \sec^2 x + 7 \right) dx \)
\( \int e^x\, dx = e^x \), \( \int \sec^2 x\, dx = \tan x \) (keep the minus sign), and \( \int 7\, dx = 7x \). The result is \( e^x - \tan x + 7x + C \). Choice (a) is missing \( +C \). Choice (b) has the wrong sign on \( \tan x \).