Introduction to :

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).

Practice:

Example 1

Find .


Example 2


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.


Example 6

Find .


Example 7

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.


Exercise 11

Integrate.


Exercise 12

Integrate.


Exercise 13

Integrate.


Exercise 14

Integrate.


Exercise 15

Integrate.


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