# At a Glance - Leibniz (Fraction) Notation

To do integration by substitution using Leibniz notation, we think of the derivative function  as a fraction of infinitesimally small quantities du and dx. We change variables by manipulating these infinitesimal quantities.

The general strategy is pretty much the same as before:

• Change variables (substitute in u for some function of x).
• Integrate.
• Put the original variable back (substitute the function of x back in for u).

 Find .

 Find .

#### Example 3

 For the integral, (a) identify u and du and (b) integrate by substitution.

#### Example 4

 For the integral, (a) identify u and du and (b) integrate by substitution.

#### Example 5

 For the integral, (a) identify u and du and (b) integrate by substitution.

 Find .

 Find .

#### Example 8

 Example. Find without using the "multiplying by 1" trick.

#### Exercise 1

For the integral, (a) identify u and du and (b) integrate by substitution.

#### Exercise 2

For the integral, (a) identify u and du and (b) integrate by substitution.

#### Exercise 3

For the integral, (a) identify u and du and (b) integrate by substitution.

#### Exercise 4

For the integral, (a) identify u and du and (b) integrate by substitution.

#### Exercise 5

For the integral, (a) identify u and du and (b) integrate by substitution.

#### Exercise 6

Integrate by substitution.

#### Exercise 7

Integrate by substitution.

#### Exercise 8

Integrate by substitution.

#### Exercise 9

Integrate by substitution.

#### Exercise 10

Integrate by substitution.

Integrate.

Integrate.

Integrate.

Integrate.

Integrate.