01 Key Concepts
Antiderivative
F(x) is an antiderivative of f(x) if F'(x) = f(x). Integration finds antiderivatives.
Indefinite Integral Notation
Written as Integral of f(x) dx = F(x) + C, where C is an arbitrary constant since derivatives of constants are zero.
The Power Rule for Integration
Integral of x^n dx = x^(n+1)/(n+1) + C, for n not equal to -1.
Constant Multiple and Sum Rules
Constants factor out of integrals, and integrals of sums split into sums of integrals — mirroring the rules for derivatives.
Why '+C' Matters
Since many functions share the same derivative (they differ by a constant), every indefinite integral needs the '+C' to represent the whole family of antiderivatives.
02 Key Formulas
- Integral of x^n dx = x^(n+1)/(n+1) + C (n does not equal -1)
- Integral of k dx = kx + C
- Integral of [f(x)+g(x)] dx = Integral of f(x)dx + Integral of g(x)dx
03 Solved Examples
Example 1 Find the integral of x^3 dx.
- Apply the power rule with n=3: x^(3+1)/(3+1) = x^4/4.
- Add the constant of integration.
Answer: x^4/4 + C
Example 2 Find the integral of (4x^2 - 3x + 5) dx.
- Integrate term by term.
- Integral of 4x^2 dx = 4x^3/3.
- Integral of -3x dx = -3x^2/2.
- Integral of 5 dx = 5x.
- Combine terms and add C.
Answer: (4x^3)/3 - (3x^2)/2 + 5x + C
Example 3 Verify that F(x) = x^2 + 7 is an antiderivative of f(x) = 2x.
- Differentiate F(x): F'(x) = 2x + 0 = 2x.
- This matches f(x) = 2x exactly.
Answer: Confirmed: F(x) is a valid antiderivative
04 Practice Questions
1Find the integral of x^5 dx.
x^6/6 + C
2Find the integral of 7 dx.
7x + C
3Find the integral of (2x + 3) dx.
x^2 + 3x + C
4Find the integral of x^(-2) dx (n not equal -1, so rule applies).
-x^(-1) + C, i.e., -1/x + C
5Is F(x) = x^3 + 5 an antiderivative of f(x) = 3x^2?
Yes
📄 Integration — Downloadable Worksheet
10 questions with a full answer key. Grab the PDF to print, or try the interactive version in your browser.