Section 5.3

Substitution Rule

The Chain Rule allows us to differentiate composite functions. Substitution (u-sub) allows us to integrate them. It allows us to simplify a complex world into a simple variable .

1

The "Inside" Job

If we have an integral like , we have a composite function and its derivative sitting right next to it.

The Transformation Machine

U-Substitution is just changing lenses.

Process0/5
1. Identify inner function
2. Identify derivative
3. Substitute (Transform)
4. Integrate
5. Back-Substitute
2

Worked Example

Exponential Composite

Evaluate .

1. Choose u

The exponent is the inner function.

2. Find du

Hey look! We have exactly and in the integral!

3. Substitute & Integrate


3

Level Up Examples

Example A: Trig Composite

Evaluate .

1. Inner Function:
.
2. Derivative:
. (Perfect mach!)
3. Result:
.

Example B: Power Composite

Evaluate .

1. Inner Function:
.
2. Derivative:
. (Perfect match!)
3. Substitute & Integrate:
.
5

Practice Quiz

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