01 Key Concepts
Family of Functions
Integral of f(x) dx = F(x) + C represents infinitely many functions, one for each possible value of C.
Geometric View
All antiderivatives of the same function are vertical shifts of one another — they share identical slopes at every x.
Finding a Particular Antiderivative
If given an initial condition (like F(0) = 5), substitute it into F(x) + C to solve for the specific value of C.
Basic Integral Rules Recap
Power rule, constant multiple rule, and sum rule all apply the same way as with definite integrals, just without evaluating at bounds.
Common Integral Forms
Integral of e^x dx = e^x + C. Integral of 1/x dx = ln|x| + C. Integral of sin(x) dx = -cos(x) + C. Integral of cos(x) dx = sin(x) + C.
02 Key Formulas
- Integral of e^x dx = e^x + C
- Integral of 1/x dx = ln|x| + C
- Integral of sin(x) dx = -cos(x) + C
- Integral of cos(x) dx = sin(x) + C
03 Solved Examples
Example 1 Find the indefinite integral of (6x^2 + 4) dx.
- Integrate term by term: Integral of 6x^2 dx = 2x^3.
- Integral of 4 dx = 4x.
- Combine and add C.
Answer: 2x^3 + 4x + C
Example 2 Find F(x) if F'(x) = 3x^2 and F(1) = 5.
- Integrate: F(x) = x^3 + C.
- Use the condition F(1)=5: 1 + C = 5, so C = 4.
Answer: F(x) = x^3 + 4
Example 3 Find the integral of (cos(x) + e^x) dx.
- Integral of cos(x) dx = sin(x).
- Integral of e^x dx = e^x.
- Combine and add C.
Answer: sin(x) + e^x + C
04 Practice Questions
1Find the integral of (10x + 2) dx.
5x^2 + 2x + C
2Find F(x) if F'(x)=2x and F(0)=3.
F(x) = x^2 + 3
3Find the integral of sin(x) dx.
-cos(x) + C
4Find the integral of (1/x) dx.
ln|x| + C
5Find the integral of 5e^x dx.
5e^x + C
📄 Indefinite Integrals — Downloadable Worksheet
10 questions with a full answer key. Grab the PDF to print, or try the interactive version in your browser.