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Integrals With Only Even Powers cos 2 xdx Integrals With Only Even - - PowerPoint PPT Presentation

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Integrals With Only Even Powers cos 2 xdx Integrals With Only Even Powers cos 2 xdx Here we need to use the trig identity: Integrals With Only Even Powers cos 2 xdx Here we need to use the trig identity: 1 + cos cos 2

slide-1
SLIDE 1
slide-2
SLIDE 2

Integrals With Only Even Powers

  • cos2 xdx
slide-3
SLIDE 3

Integrals With Only Even Powers

  • cos2 xdx

Here we need to use the trig identity:

slide-4
SLIDE 4

Integrals With Only Even Powers

  • cos2 xdx

Here we need to use the trig identity: cos θ 2 =

  • 1 + cos θ

2

slide-5
SLIDE 5

Integrals With Only Even Powers

  • cos2 xdx

Here we need to use the trig identity: cos θ 2 =

  • 1 + cos θ

2 If we make x = θ

2, we have that:

slide-6
SLIDE 6

Integrals With Only Even Powers

  • cos2 xdx

Here we need to use the trig identity: cos θ 2 =

  • 1 + cos θ

2 If we make x = θ

2, we have that:

cos x =

  • 1 + cos 2x

2

slide-7
SLIDE 7

Integrals With Only Even Powers

  • cos2 xdx

Here we need to use the trig identity: cos θ 2 =

  • 1 + cos θ

2 If we make x = θ

2, we have that:

cos x =

  • 1 + cos 2x

2 And if we square both sides:

slide-8
SLIDE 8

Integrals With Only Even Powers

  • cos2 xdx

Here we need to use the trig identity: cos θ 2 =

  • 1 + cos θ

2 If we make x = θ

2, we have that:

cos x =

  • 1 + cos 2x

2 And if we square both sides: cos2 x = 1 + cos 2x 2

slide-9
SLIDE 9

Integrals With Only Even Powers

So, we replace in our integral:

slide-10
SLIDE 10

Integrals With Only Even Powers

So, we replace in our integral: cos2 x = 1 + cos 2x 2

slide-11
SLIDE 11

Integrals With Only Even Powers

So, we replace in our integral: cos2 x = 1 + cos 2x 2

  • cos2 xdx =
slide-12
SLIDE 12

Integrals With Only Even Powers

So, we replace in our integral: cos2 x = 1 + cos 2x 2

  • cos2 xdx =
slide-13
SLIDE 13

Integrals With Only Even Powers

So, we replace in our integral: cos2 x = 1 + cos 2x 2

  • cos2 xdx =

1 + cos 2x 2

slide-14
SLIDE 14

Integrals With Only Even Powers

So, we replace in our integral: cos2 x = 1 + cos 2x 2

  • cos2 xdx =

1 + cos 2x 2 dx

slide-15
SLIDE 15

Integrals With Only Even Powers

So, we replace in our integral: cos2 x = 1 + cos 2x 2

  • cos2 xdx =

1 + cos 2x 2 dx =

slide-16
SLIDE 16

Integrals With Only Even Powers

So, we replace in our integral: cos2 x = 1 + cos 2x 2

  • cos2 xdx =

1 + cos 2x 2 dx = 1 2 + cos 2x 2

  • dx
slide-17
SLIDE 17

Integrals With Only Even Powers

So, we replace in our integral: cos2 x = 1 + cos 2x 2

  • cos2 xdx =

1 + cos 2x 2 dx = 1 2 + cos 2x 2

  • dx

=

slide-18
SLIDE 18

Integrals With Only Even Powers

So, we replace in our integral: cos2 x = 1 + cos 2x 2

  • cos2 xdx =

1 + cos 2x 2 dx = 1 2 + cos 2x 2

  • dx

= 1 2

slide-19
SLIDE 19

Integrals With Only Even Powers

So, we replace in our integral: cos2 x = 1 + cos 2x 2

  • cos2 xdx =

1 + cos 2x 2 dx = 1 2 + cos 2x 2

  • dx

= 1 2dx+

slide-20
SLIDE 20

Integrals With Only Even Powers

So, we replace in our integral: cos2 x = 1 + cos 2x 2

  • cos2 xdx =

1 + cos 2x 2 dx = 1 2 + cos 2x 2

  • dx

= 1 2dx +

slide-21
SLIDE 21

Integrals With Only Even Powers

So, we replace in our integral: cos2 x = 1 + cos 2x 2

  • cos2 xdx =

1 + cos 2x 2 dx = 1 2 + cos 2x 2

  • dx

= 1 2dx + cos 2x 2

slide-22
SLIDE 22

Integrals With Only Even Powers

So, we replace in our integral: cos2 x = 1 + cos 2x 2

  • cos2 xdx =

1 + cos 2x 2 dx = 1 2 + cos 2x 2

  • dx

= 1 2dx + cos 2x 2 dx

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SLIDE 23

Integrals With Only Even Powers

So, we replace in our integral: cos2 x = 1 + cos 2x 2

  • cos2 xdx =

1 + cos 2x 2 dx = 1 2 + cos 2x 2

  • dx

= 1 2dx + cos 2x 2 dx = 1 2

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SLIDE 24

Integrals With Only Even Powers

So, we replace in our integral: cos2 x = 1 + cos 2x 2

  • cos2 xdx =

1 + cos 2x 2 dx = 1 2 + cos 2x 2

  • dx

= 1 2dx + cos 2x 2 dx = 1 2

slide-25
SLIDE 25

Integrals With Only Even Powers

So, we replace in our integral: cos2 x = 1 + cos 2x 2

  • cos2 xdx =

1 + cos 2x 2 dx = 1 2 + cos 2x 2

  • dx

= 1 2dx + cos 2x 2 dx = 1 2

  • dx+
slide-26
SLIDE 26

Integrals With Only Even Powers

So, we replace in our integral: cos2 x = 1 + cos 2x 2

  • cos2 xdx =

1 + cos 2x 2 dx = 1 2 + cos 2x 2

  • dx

= 1 2dx + cos 2x 2 dx = 1 2

  • dx + 1

2

slide-27
SLIDE 27

Integrals With Only Even Powers

So, we replace in our integral: cos2 x = 1 + cos 2x 2

  • cos2 xdx =

1 + cos 2x 2 dx = 1 2 + cos 2x 2

  • dx

= 1 2dx + cos 2x 2 dx = 1 2

  • dx + 1

2

slide-28
SLIDE 28

Integrals With Only Even Powers

So, we replace in our integral: cos2 x = 1 + cos 2x 2

  • cos2 xdx =

1 + cos 2x 2 dx = 1 2 + cos 2x 2

  • dx

= 1 2dx + cos 2x 2 dx = 1 2

  • dx + 1

2

  • cos 2x
slide-29
SLIDE 29

Integrals With Only Even Powers

So, we replace in our integral: cos2 x = 1 + cos 2x 2

  • cos2 xdx =

1 + cos 2x 2 dx = 1 2 + cos 2x 2

  • dx

= 1 2dx + cos 2x 2 dx = 1 2

  • dx + 1

2

  • cos 2xdx
slide-30
SLIDE 30

Integrals With Only Even Powers

So, we replace in our integral: cos2 x = 1 + cos 2x 2

  • cos2 xdx =

1 + cos 2x 2 dx = 1 2 + cos 2x 2

  • dx

= 1 2dx + cos 2x 2 dx = 1 2

  • dx

x

+1 2

  • cos 2xdx
slide-31
SLIDE 31

Integrals With Only Even Powers

So, we replace in our integral: cos2 x = 1 + cos 2x 2

  • cos2 xdx =

1 + cos 2x 2 dx = 1 2 + cos 2x 2

  • dx

= 1 2dx + cos 2x 2 dx = 1 2

  • dx

x

+1 2

  • cos 2xdx

= x 2+

slide-32
SLIDE 32

Integrals With Only Even Powers

So, we replace in our integral: cos2 x = 1 + cos 2x 2

  • cos2 xdx =

1 + cos 2x 2 dx = 1 2 + cos 2x 2

  • dx

= 1 2dx + cos 2x 2 dx = 1 2

  • dx

x

+1 2

  • cos 2xdx

= x 2 + 1 2

  • cos 2xdx
slide-33
SLIDE 33

Integrals With Only Even Powers

x 2 + 1 2

  • cos 2xdx
slide-34
SLIDE 34

Integrals With Only Even Powers

x 2 + 1 2

  • cos 2xdx

We solve this integral by substitution:

slide-35
SLIDE 35

Integrals With Only Even Powers

x 2 + 1 2

  • cos 2xdx

We solve this integral by substitution: u =

slide-36
SLIDE 36

Integrals With Only Even Powers

x 2 + 1 2

  • cos 2xdx

We solve this integral by substitution: u = 2x

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SLIDE 37

Integrals With Only Even Powers

x 2 + 1 2

  • cos 2xdx

We solve this integral by substitution: u = 2x,

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SLIDE 38

Integrals With Only Even Powers

x 2 + 1 2

  • cos 2xdx

We solve this integral by substitution: u = 2x, du dx =

slide-39
SLIDE 39

Integrals With Only Even Powers

x 2 + 1 2

  • cos 2xdx

We solve this integral by substitution: u = 2x, du dx = 2

slide-40
SLIDE 40

Integrals With Only Even Powers

x 2 + 1 2

  • cos 2xdx

We solve this integral by substitution: u = 2x, du dx = 2 x 2+

slide-41
SLIDE 41

Integrals With Only Even Powers

x 2 + 1 2

  • cos 2xdx

We solve this integral by substitution: u = 2x, du dx = 2 x 2 + 1 2

slide-42
SLIDE 42

Integrals With Only Even Powers

x 2 + 1 2

  • cos 2xdx

We solve this integral by substitution: u = 2x, du dx = 2 x 2 + 1 2

slide-43
SLIDE 43

Integrals With Only Even Powers

x 2 + 1 2

  • cos 2xdx

We solve this integral by substitution: u = 2x, du dx = 2 x 2 + 1 2

  • cos u
slide-44
SLIDE 44

Integrals With Only Even Powers

x 2 + 1 2

  • cos 2xdx

We solve this integral by substitution: u = 2x, du dx = 2 x 2 + 1 2

  • cos u.

1 2 du dx

slide-45
SLIDE 45

Integrals With Only Even Powers

x 2 + 1 2

  • cos 2xdx

We solve this integral by substitution: u = 2x, du dx = 2 x 2 + 1 2

  • cos u.

1 2 du dx

  • dx
slide-46
SLIDE 46

Integrals With Only Even Powers

x 2 + 1 2

  • cos 2xdx

We solve this integral by substitution: u = 2x, du dx = 2 x 2 + 1 2

  • cos u.

1 2 du

✚ ✚

dx

  • ✚

✚

dx

slide-47
SLIDE 47

Integrals With Only Even Powers

x 2 + 1 2

  • cos 2xdx

We solve this integral by substitution: u = 2x, du dx = 2 x 2 + 1 2

  • cos u.

1 2 du

✚ ✚

dx

  • ✚

✚

dx = x 2+

slide-48
SLIDE 48

Integrals With Only Even Powers

x 2 + 1 2

  • cos 2xdx

We solve this integral by substitution: u = 2x, du dx = 2 x 2 + 1 2

  • cos u.

1 2 du

✚ ✚

dx

  • ✚

✚

dx = x 2 + 1 4

slide-49
SLIDE 49

Integrals With Only Even Powers

x 2 + 1 2

  • cos 2xdx

We solve this integral by substitution: u = 2x, du dx = 2 x 2 + 1 2

  • cos u.

1 2 du

✚ ✚

dx

  • ✚

✚

dx = x 2 + 1 4

slide-50
SLIDE 50

Integrals With Only Even Powers

x 2 + 1 2

  • cos 2xdx

We solve this integral by substitution: u = 2x, du dx = 2 x 2 + 1 2

  • cos u.

1 2 du

✚ ✚

dx

  • ✚

✚

dx = x 2 + 1 4

  • cos u
slide-51
SLIDE 51

Integrals With Only Even Powers

x 2 + 1 2

  • cos 2xdx

We solve this integral by substitution: u = 2x, du dx = 2 x 2 + 1 2

  • cos u.

1 2 du

✚ ✚

dx

  • ✚

✚

dx = x 2 + 1 4

  • cos udu
slide-52
SLIDE 52

Integrals With Only Even Powers

x 2 + 1 2

  • cos 2xdx

We solve this integral by substitution: u = 2x, du dx = 2 x 2 + 1 2

  • cos u.

1 2 du

✚ ✚

dx

  • ✚

✚

dx = x 2 + 1 4

  • cos udu
  • sin u
slide-53
SLIDE 53

Integrals With Only Even Powers

x 2 + 1 2

  • cos 2xdx

We solve this integral by substitution: u = 2x, du dx = 2 x 2 + 1 2

  • cos u.

1 2 du

✚ ✚

dx

  • ✚

✚

dx = x 2 + 1 4

  • cos udu = x

2

slide-54
SLIDE 54

Integrals With Only Even Powers

x 2 + 1 2

  • cos 2xdx

We solve this integral by substitution: u = 2x, du dx = 2 x 2 + 1 2

  • cos u.

1 2 du

✚ ✚

dx

  • ✚

✚

dx = x 2 + 1 4

  • cos udu = x

2

slide-55
SLIDE 55

Integrals With Only Even Powers

x 2 + 1 2

  • cos 2xdx

We solve this integral by substitution: u = 2x, du dx = 2 x 2 + 1 2

  • cos u.

1 2 du

✚ ✚

dx

  • ✚

✚

dx = x 2 + 1 4

  • cos udu = x

2+

slide-56
SLIDE 56

Integrals With Only Even Powers

x 2 + 1 2

  • cos 2xdx

We solve this integral by substitution: u = 2x, du dx = 2 x 2 + 1 2

  • cos u.

1 2 du

✚ ✚

dx

  • ✚

✚

dx = x 2 + 1 4

  • cos udu = x

2 + 1 4

slide-57
SLIDE 57

Integrals With Only Even Powers

x 2 + 1 2

  • cos 2xdx

We solve this integral by substitution: u = 2x, du dx = 2 x 2 + 1 2

  • cos u.

1 2 du

✚ ✚

dx

  • ✚

✚

dx = x 2 + 1 4

  • cos udu = x

2 + 1 4 sin u + C

slide-58
SLIDE 58

Integrals With Only Even Powers

x 2 + 1 2

  • cos 2xdx

We solve this integral by substitution: u = 2x, du dx = 2 x 2 + 1 2

  • cos u.

1 2 du

✚ ✚

dx

  • ✚

✚

dx = x 2 + 1 4

  • cos udu = x

2 + 1 4 sin u + C

slide-59
SLIDE 59

Integrals With Only Even Powers

x 2 + 1 2

  • cos 2xdx

We solve this integral by substitution: u = 2x, du dx = 2 x 2 + 1 2

  • cos u.

1 2 du

✚ ✚

dx

  • ✚

✚

dx = x 2 + 1 4

  • cos udu = x

2 + 1 4 sin u + C

  • cos2 x
slide-60
SLIDE 60

Integrals With Only Even Powers

x 2 + 1 2

  • cos 2xdx

We solve this integral by substitution: u = 2x, du dx = 2 x 2 + 1 2

  • cos u.

1 2 du

✚ ✚

dx

  • ✚

✚

dx = x 2 + 1 4

  • cos udu = x

2 + 1 4 sin u + C

  • cos2 xdx
slide-61
SLIDE 61

Integrals With Only Even Powers

x 2 + 1 2

  • cos 2xdx

We solve this integral by substitution: u = 2x, du dx = 2 x 2 + 1 2

  • cos u.

1 2 du

✚ ✚

dx

  • ✚

✚

dx = x 2 + 1 4

  • cos udu = x

2 + 1 4 sin u + C

  • cos2 xdx =
slide-62
SLIDE 62

Integrals With Only Even Powers

x 2 + 1 2

  • cos 2xdx

We solve this integral by substitution: u = 2x, du dx = 2 x 2 + 1 2

  • cos u.

1 2 du

✚ ✚

dx

  • ✚

✚

dx = x 2 + 1 4

  • cos udu = x

2 + 1 4 sin u + C

  • cos2 xdx = x

2

slide-63
SLIDE 63

Integrals With Only Even Powers

x 2 + 1 2

  • cos 2xdx

We solve this integral by substitution: u = 2x, du dx = 2 x 2 + 1 2

  • cos u.

1 2 du

✚ ✚

dx

  • ✚

✚

dx = x 2 + 1 4

  • cos udu = x

2 + 1 4 sin u + C

  • cos2 xdx = x

2+

slide-64
SLIDE 64

Integrals With Only Even Powers

x 2 + 1 2

  • cos 2xdx

We solve this integral by substitution: u = 2x, du dx = 2 x 2 + 1 2

  • cos u.

1 2 du

✚ ✚

dx

  • ✚

✚

dx = x 2 + 1 4

  • cos udu = x

2 + 1 4 sin u + C

  • cos2 xdx = x

2 + 1 4

slide-65
SLIDE 65

Integrals With Only Even Powers

x 2 + 1 2

  • cos 2xdx

We solve this integral by substitution: u = 2x, du dx = 2 x 2 + 1 2

  • cos u.

1 2 du

✚ ✚

dx

  • ✚

✚

dx = x 2 + 1 4

  • cos udu = x

2 + 1 4 sin u + C

  • cos2 xdx = x

2 + 1 4 sin 2x

slide-66
SLIDE 66

Integrals With Only Even Powers

x 2 + 1 2

  • cos 2xdx

We solve this integral by substitution: u = 2x, du dx = 2 x 2 + 1 2

  • cos u.

1 2 du

✚ ✚

dx

  • ✚

✚

dx = x 2 + 1 4

  • cos udu = x

2 + 1 4 sin u + C

  • cos2 xdx = x

2 + 1 4 sin 2x + C

slide-67
SLIDE 67

Integrals With Only Even Powers

x 2 + 1 2

  • cos 2xdx

We solve this integral by substitution: u = 2x, du dx = 2 x 2 + 1 2

  • cos u.

1 2 du

✚ ✚

dx

  • ✚

✚

dx = x 2 + 1 4

  • cos udu = x

2 + 1 4 sin u + C

  • cos2 xdx = x

2 + 1 4 sin 2x + C

slide-68
SLIDE 68