Lesson 5 5c
This presentation is the property of its rightful owner.
Sponsored Links
1 / 11

Lesson 5-5c PowerPoint PPT Presentation


  • 54 Views
  • Uploaded on
  • Presentation posted in: General

Lesson 5-5c. U-Substitution or The Chain Rule of Integration. Quiz. ∫. cos (3x) dx =. Homework Problem: Reading questions: Which is even, which is odd?. Objectives. Recognize when to try ‘u’ substitution techniques

Download Presentation

Lesson 5-5c

An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -

Presentation Transcript


Lesson 5 5c

Lesson 5-5c

U-Substitution or The Chain Rule of Integration


Lesson 5 5c

Quiz

cos (3x) dx =

  • Homework Problem:

  • Reading questions: Which is even, which is odd?


Objectives

Objectives

  • Recognize when to try ‘u’ substitution techniques

  • Solve integrals of algebraic, exponential, logarithmic, and trigonometric functions using ‘u’ substitution technique

  • Use symmetry to solve integrals about x = 0 (y-axis)


Vocabulary

Vocabulary

  • Change of Variable – substitution of one variable for another in an integral (sort of reverse of the chain rule)

  • Even Functions – when f(-x) = f(x); even functions are symmetric to the y-axis

  • Odd Functions – when f(-x) = -f(x); odd functions are symmetric to the origin


Integrals of symmetric functions

Integrals of Symmetric Functions

  • If a function, f(x), is even [f(-x) = f(x)], then its integral from –a to a is

  • If a function, f(x), is even [f(-x) = -f(x)], then its integral from –a to a isbecause of signed area (above axis and below) cancel each other out.

a

a

a

-a

0

-a

f(x) dx = 2 f(x) dx

f(x) dx = 0


Example 1

Example 1

= ∫(2x + 1)² 2/2 dx

∫(2x + 1)² dx

If we let u = 2x +1then it becomes u² and du = 2dx

we are missing a 2 from dxso we multiple by 1 (2/2)

= ½ ∫(2x + 1)² 2 dx

½ goes outside ∫ and 2 stays with dx

= ½ ∫u² du = 1/6 u³ + C

= 1/6 (2x + 1)³ + C


Example problems

Example Problems

1) (2x +3) cos(x² + 3x) dx

Find the derivative of each of the following:

  • (5x² + 1)² (10x) dx

Let u = x² + 3x

then du = 2x + 3

So it becomes

cos u du

Let u = 5x² + 1

then du = 10x

So it becomes

u² du

= cos u du = sin (u) + C

= u² du = ⅓ u³ + C

= sin (x² + 3x) + C

= ⅓ (5x² + 1)³ + C


Example problems cont

Example Problems cont

4

3) tan²(t) cot (t) dt

Find the derivative of each of the following:

π/4

x

t tan (t)dt

x


Example problems cont1

6) √2 + sin (t) dt

2

Example Problems cont

2x

5) 5t sin (t) dt

Find the derivative of each of the following:

1


Example problems cont2

Example Problems cont

7) √1 + t4dt

Find the derivative of each of the following:

cos x

x

8) t² dt

sin x


Summary homework

Summary & Homework

  • Summary:

    • alksdfj

  • Homework:

    • Day One: pg 420 - 422: 1, 2, 6, 8, 13, 21, 35, 42, 51, 58, 59, 76

    • Day Two: pg 420 - 422: 1, 2, 6, 8, 13, 21, 35, 42, 51, 58, 59, 76


  • Login