Introduction to Integration
Discover integration as the reverse process of differentiation, understand anti-derivatives, and learn the foundational notation and concepts.
What is Integration?
Integration is the reverse process of differentiation. If differentiation asks "What is the rate of change?", integration asks "What function has this rate of change?" This reverse process is also called anti-differentiation.
If d/dx[F(x)] = f(x), then ∫ f(x) dx = F(x) + C
F(x) is an anti-derivative (or primitive) of f(x). C is the constant of integration.
For example, since d/dx(x3) = 3x2, we know that ∫ 3x2 dx = x3 + C.
Differentiation vs Integration
Differentiation
x3 → 3x2
"Find the derivative"
Integration
3x2 → x3 + C
"Find the anti-derivative"
Integration Notation
The integral sign ∫ is an elongated "S" (for "sum"). The complete notation has several parts:
∫ = the integral sign (tells us to integrate)
f(x) = the integrand (the function to be integrated)
dx = indicates we are integrating with respect to x
Why +C? When we differentiate a constant, it vanishes (d/dx[5] = 0). So when reversing the process, we must add an arbitrary constant C because we do not know what constant was there originally.
The Power Rule for Integration
The most fundamental integration rule reverses the power rule for differentiation:
∫ xn dx = xn+1n + 1 + C (n ≠ −1)
Increase the power by 1, then divide by the new power.
Quick Examples
∫ x4 dx = x5/5 + C
∫ x dx = x2/2 + C
∫ 1 dx = x + C
∫ x−2 dx = x−1/(−1) + C = −1/x + C
Key Vocabulary
Integration
The reverse of differentiation; finding a function whose derivative is the given function.
Anti-derivative
A function F(x) such that F'(x) = f(x). Also called a primitive function.
Integrand
The function being integrated, written between the ∫ sign and dx.
Constant of Integration
The +C added to every indefinite integral, representing an unknown constant.
Worked Examples
Find ∫ 6x2 dx.
Step 1: Apply the power rule to x2: ∫ x2 dx = x3/3.
Step 2: Multiply by the constant 6: 6 × x3/3 = 2x3.
Answer: ∫ 6x2 dx = 2x3 + C
Find ∫ (3x2 + 4x − 5) dx.
Step 1: Integrate each term separately.
Step 2: ∫ 3x2 dx = x3, ∫ 4x dx = 2x2, ∫ −5 dx = −5x.
Answer: x3 + 2x2 − 5x + C
Find the anti-derivative of f(x) = 1/x3.
Step 1: Rewrite: 1/x3 = x−3.
Step 2: Apply power rule: ∫ x−3 dx = x−2/(−2) = −1/(2x2).
Answer: −1/(2x2) + C
Knowledge Check
Select the correct answer for each question. Click "Check Answer" to see if you are right.
Question 1
What is ∫ x5 dx?
Question 2
What is ∫ 4 dx?
Question 3
Why do we include "+ C" in an indefinite integral?
Question 4
What is ∫ (2x + 3) dx?
Question 5
What is ∫ √x dx? (Hint: rewrite √x as x1/2)
Key Concepts Summary
- ● Integration is the reverse of differentiation (anti-differentiation).
- ● The power rule for integration: ∫ xn dx = xn+1/(n+1) + C, where n ≠ −1.
- ● Always include the constant of integration +C for indefinite integrals.
- ● You can verify an integral by differentiating the answer — you should get back the original integrand.
- ● Integrals of sums/differences can be computed term by term.