We have changed our privacy policy. In addition, we use cookies on our website for various purposes. By continuing on our website, you consent to our use of cookies. You can learn about our practices by reading our privacy policy.
© 2016 Shmoop University, Inc. All rights reserved.
GO TO SAT PREP GO TO ACT PREP

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).

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.


People who Shmooped this also Shmooped...

Advertisement