Section 5.4

Advanced Substitution

Sometimes the perfect substitution isn't handed to us. We have to manipulate constants or rearrange terms to make the pieces fit.

1

Constants and Balance

If substitution requires a constant (like or ) that isn't there, we can introduce it by multiplying the inside by and the outside by .

The Balance Scale

Balancing the Differential

Let
Then
We have . We need .
1.00
2.00
Needed for du
Outer Weight1

Adjust the outer constant to neutralize the inner 2.

2

Worked Example: Back Substitution

Dealing with Leftovers

Evaluate .

1. Choose u

Problem: We have an extra outside!

2. Solve for x

Since , then .

3. Substitute & Expand


3

Level Up Examples

Example A: The Hidden Log

Evaluate .

1. Rewrite:
.
2. Substitute:
.
So .
3. Result:
.

Example B: Rational Back-Sub

Evaluate .

1. Setup:
..
2. Transform:

.
3. Result:
.
5

Practice Quiz

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