Integration

Main Content

What is Integration?

Integration is the opposite (or reverse) of differentiation. If differentiation tells us the rate of change of a function, integration does the opposite - it finds the original function when we know its rate of change.

Think of it this way:

• Differentiation takes you from a function to its derivative (gradient)
• Integration takes you from a derivative back to the original function

For example:

• If we differentiate f(x) = x² + 3, we get f'(x) = 2x
• If we integrate 2x, we get back to x² + 3 (plus a constant - more on this later)

The symbol for integration is ∫ (called an integral sign). When we write ∫ 2x dx, it means "integrate 2x with respect to x".

The dx at the end tells us which variable we're integrating with respect to (in this case, x).

The Basic Rule for Integration (Power Rule)

When we integrate x^n (x raised to a power), we use this rule:

∫ x^n dx = (x^(n+1))/(n+1) + c

where n ≠ -1 (we can't use this rule when n = -1)

Step-by-step process:

1. Add 1 to the power
2. Divide by the new power
3. Add a constant c (called the constant of integration)

Why do we add c? When we differentiate, any constant becomes zero. So when we integrate, we don't know what constant was there originally - it could have been any number. We write "+ c" to represent this unknown constant.

Examples:

Example 1: Integrate x³

∫ x³ dx = x^(3+1)/(3+1) + c = x⁴/4 + c

Example 2: Integrate x⁵

∫ x⁵ dx = x^(5+1)/(5+1) + c = x⁶/6 + c

Example 3: Integrate x^(-2) (the same as 1/x²)

∫ x^(-2) dx = x^(-2+1)/(-2+1) + c = x^(-1)/(-1) + c = -1/x + c

Integrating (ax + b)^n

Sometimes we need to integrate expressions like (2x + 3)^4 or (5x - 1)^(-2). These are in the form (ax + b)^n where:

• a and b are constants (numbers)
• n is any rational number except -1

The formula is:

∫ (ax + b)^n dx = (ax + b)^(n+1) / [a(n+1)] + c

Step-by-step process:

1. Add 1 to the power (n becomes n+1)
2. Divide by the new power (n+1)
3. Divide by the coefficient of x (which is a)
4. Add the constant c

Examples:

Example 4: Integrate (2x + 3)⁴

∫ (2x + 3)⁴ dx = (2x + 3)^(4+1) / [2(4+1)] + c = (2x + 3)⁵/10 + c

Here: a = 2, b = 3, n = 4

Example 5: Integrate 1/(2x + 3)²

First, rewrite this as (2x + 3)^(-2)

∫ (2x + 3)^(-2) dx = (2x + 3)^(-2+1) / [2(-2+1)] + c = (2x + 3)^(-1) / [2(-1)] + c = (2x + 3)^(-1) / (-2) + c

Simplifying: = -1/[2(2x + 3)] + c

Example 6: Integrate √(3x - 1)

First, write the square root as a power: √(3x - 1) = (3x - 1)^(1/2)

∫ (3x - 1)^(1/2) dx = (3x - 1)^(1/2 + 1) / [3(1/2 + 1)] + c = (3x - 1)^(3/2) / [3(3/2)] + c = (3x - 1)^(3/2) / (9/2) + c

Simplifying: = 2(3x - 1)^(3/2) / 9 + c