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Trigonometric Integrals Trigonometric Integrals There are four - - PowerPoint PPT Presentation

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Trigonometric Integrals Trigonometric Integrals There are four cases: Trigonometric Integrals There are four cases: 1. Odd power of sin x or cos x : Trigonometric Integrals There are four cases: 1. Odd power of sin x or cos x : sin 3 xdx

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SLIDE 1
slide-2
SLIDE 2

Trigonometric Integrals

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SLIDE 3

Trigonometric Integrals

There are four cases:

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SLIDE 4

Trigonometric Integrals

There are four cases:

  • 1. Odd power of sin x or cos x:
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SLIDE 5

Trigonometric Integrals

There are four cases:

  • 1. Odd power of sin x or cos x:
  • sin3 xdx
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SLIDE 6

Trigonometric Integrals

There are four cases:

  • 1. Odd power of sin x or cos x:
  • sin3 xdx
  • 2. Even power of sin x or cos x:
slide-7
SLIDE 7

Trigonometric Integrals

There are four cases:

  • 1. Odd power of sin x or cos x:
  • sin3 xdx
  • 2. Even power of sin x or cos x:
  • cos2 xdx
slide-8
SLIDE 8

Trigonometric Integrals

There are four cases:

  • 1. Odd power of sin x or cos x:
  • sin3 xdx
  • 2. Even power of sin x or cos x:
  • cos2 xdx
  • 3. Integrals involving tan x and sec x:
slide-9
SLIDE 9

Trigonometric Integrals

There are four cases:

  • 1. Odd power of sin x or cos x:
  • sin3 xdx
  • 2. Even power of sin x or cos x:
  • cos2 xdx
  • 3. Integrals involving tan x and sec x:
  • tan3 xdx
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SLIDE 10

Trigonometric Integrals

There are four cases:

  • 1. Odd power of sin x or cos x:
  • sin3 xdx
  • 2. Even power of sin x or cos x:
  • cos2 xdx
  • 3. Integrals involving tan x and sec x:
  • tan3 xdx
  • 4. Just tricky trig integrals (use any trig identity you know):
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SLIDE 11

Trigonometric Integrals

There are four cases:

  • 1. Odd power of sin x or cos x:
  • sin3 xdx
  • 2. Even power of sin x or cos x:
  • cos2 xdx
  • 3. Integrals involving tan x and sec x:
  • tan3 xdx
  • 4. Just tricky trig integrals (use any trig identity you know):
  • sin 2xdx

cos4 x + sin4 x

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SLIDE 12

Odd Powers of cos x and sin x

  • sin3 xdx
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SLIDE 13

Odd Powers of cos x and sin x

  • sin3 xdx

We can write this integral as:

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SLIDE 14

Odd Powers of cos x and sin x

  • sin3 xdx

We can write this integral as:

  • sin2 x sin xdx
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SLIDE 15

Odd Powers of cos x and sin x

  • sin3 xdx

We can write this integral as:

  • sin2 x sin xdx

Let’s use the identity:

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SLIDE 16

Odd Powers of cos x and sin x

  • sin3 xdx

We can write this integral as:

  • sin2 x sin xdx

Let’s use the identity: sin2 x = 1 − cos2 x

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SLIDE 17

Odd Powers of cos x and sin x

  • sin3 xdx

We can write this integral as:

  • sin2 x sin xdx

Let’s use the identity: sin2 x = 1 − cos2 x

  • sin2 x sin xdx =
slide-18
SLIDE 18

Odd Powers of cos x and sin x

  • sin3 xdx

We can write this integral as:

  • sin2 x sin xdx

Let’s use the identity: sin2 x = 1 − cos2 x

  • sin2 x sin xdx =
slide-19
SLIDE 19

Odd Powers of cos x and sin x

  • sin3 xdx

We can write this integral as:

  • sin2 x sin xdx

Let’s use the identity: sin2 x = 1 − cos2 x

  • sin2 x sin xdx =

1 − cos2 x

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SLIDE 20

Odd Powers of cos x and sin x

  • sin3 xdx

We can write this integral as:

  • sin2 x sin xdx

Let’s use the identity: sin2 x = 1 − cos2 x

  • sin2 x sin xdx =

1 − cos2 x

  • sin xdx
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SLIDE 21

Odd Powers of cos x and sin x

  • sin3 xdx

We can write this integral as:

  • sin2 x sin xdx

Let’s use the identity: sin2 x = 1 − cos2 x

  • sin2 x sin xdx =

1 − cos2 x

  • sin xdx

= sin x − cos2 x sin x

  • dx
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SLIDE 22

Odd Powers of cos x and sin x

  • sin3 xdx

We can write this integral as:

  • sin2 x sin xdx

Let’s use the identity: sin2 x = 1 − cos2 x

  • sin2 x sin xdx =

1 − cos2 x

  • sin xdx

= sin x − cos2 x sin x

  • dx =
  • sin xdx−
slide-23
SLIDE 23

Odd Powers of cos x and sin x

  • sin3 xdx

We can write this integral as:

  • sin2 x sin xdx

Let’s use the identity: sin2 x = 1 − cos2 x

  • sin2 x sin xdx =

1 − cos2 x

  • sin xdx

= sin x − cos2 x sin x

  • dx =
  • sin xdx −
  • cos2 x sin xdx
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SLIDE 24

Odd Powers of cos x and sin x

  • sin xdx −
  • cos2 x sin xdx =
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SLIDE 25

Odd Powers of cos x and sin x

✟✟✟✟✟ ✯− cos x

  • sin xdx −
  • cos2 x sin xdx =
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SLIDE 26

Odd Powers of cos x and sin x

✟✟✟✟✟ ✯− cos x

  • sin xdx −
  • cos2 x sin xdx = − cos x−
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SLIDE 27

Odd Powers of cos x and sin x

✟✟✟✟✟ ✯− cos x

  • sin xdx −
  • cos2 x sin xdx = − cos x −
  • cos2 x sin xdx
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SLIDE 28

Odd Powers of cos x and sin x

✟✟✟✟✟ ✯− cos x

  • sin xdx −
  • cos2 x sin xdx = − cos x −
  • cos2 x sin xdx

Now, we solve this integral by substitution:

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SLIDE 29

Odd Powers of cos x and sin x

✟✟✟✟✟ ✯− cos x

  • sin xdx −
  • cos2 x sin xdx = − cos x −
  • cos2 x sin xdx

Now, we solve this integral by substitution: u = cos x,

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SLIDE 30

Odd Powers of cos x and sin x

✟✟✟✟✟ ✯− cos x

  • sin xdx −
  • cos2 x sin xdx = − cos x −
  • cos2 x sin xdx

Now, we solve this integral by substitution: u = cos x, du dx = − sin x

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SLIDE 31

Odd Powers of cos x and sin x

✟✟✟✟✟ ✯− cos x

  • sin xdx −
  • cos2 x sin xdx = − cos x −
  • cos2 x sin xdx

Now, we solve this integral by substitution: u = cos x, du dx = − sin x − cos x −

  • cos2 x sin xdx =
slide-32
SLIDE 32

Odd Powers of cos x and sin x

✟✟✟✟✟ ✯− cos x

  • sin xdx −
  • cos2 x sin xdx = − cos x −
  • cos2 x sin xdx

Now, we solve this integral by substitution: u = cos x, du dx = − sin x − cos x −

  • cos2 x sin xdx = − cos x −
slide-33
SLIDE 33

Odd Powers of cos x and sin x

✟✟✟✟✟ ✯− cos x

  • sin xdx −
  • cos2 x sin xdx = − cos x −
  • cos2 x sin xdx

Now, we solve this integral by substitution: u = cos x, du dx = − sin x − cos x −

  • cos2 x sin xdx = − cos x −
  • u2
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SLIDE 34

Odd Powers of cos x and sin x

✟✟✟✟✟ ✯− cos x

  • sin xdx −
  • cos2 x sin xdx = − cos x −
  • cos2 x sin xdx

Now, we solve this integral by substitution: u = cos x, du dx = − sin x − cos x −

  • cos2 x sin xdx = − cos x −
  • u2
  • −du

dx

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SLIDE 35

Odd Powers of cos x and sin x

✟✟✟✟✟ ✯− cos x

  • sin xdx −
  • cos2 x sin xdx = − cos x −
  • cos2 x sin xdx

Now, we solve this integral by substitution: u = cos x, du dx = − sin x − cos x −

  • cos2 x sin xdx = − cos x −
  • u2
  • −du

dx

  • dx
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SLIDE 36

Odd Powers of cos x and sin x

✟✟✟✟✟ ✯− cos x

  • sin xdx −
  • cos2 x sin xdx = − cos x −
  • cos2 x sin xdx

Now, we solve this integral by substitution: u = cos x, du dx = − sin x − cos x −

  • cos2 x sin xdx = − cos x −
  • u2
  • −du

✚ ✚

dx

  • ✚

✚

dx

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SLIDE 37

Odd Powers of cos x and sin x

✟✟✟✟✟ ✯− cos x

  • sin xdx −
  • cos2 x sin xdx = − cos x −
  • cos2 x sin xdx

Now, we solve this integral by substitution: u = cos x, du dx = − sin x − cos x −

  • cos2 x sin xdx = − cos x
  • −
  • u2
  • −du

✚ ✚

dx

  • ✚

✚

dx

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SLIDE 38

Odd Powers of cos x and sin x

✟✟✟✟✟ ✯− cos x

  • sin xdx −
  • cos2 x sin xdx = − cos x −
  • cos2 x sin xdx

Now, we solve this integral by substitution: u = cos x, du dx = − sin x − cos x −

  • cos2 x sin xdx = − cos x
  • −
  • u2
  • −du

✚ ✚

dx

  • ✚

✚

dx = − cos x +

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SLIDE 39

Odd Powers of cos x and sin x

✟✟✟✟✟ ✯− cos x

  • sin xdx −
  • cos2 x sin xdx = − cos x −
  • cos2 x sin xdx

Now, we solve this integral by substitution: u = cos x, du dx = − sin x − cos x −

  • cos2 x sin xdx = − cos x
  • −
  • u2
  • −du

✚ ✚

dx

  • ✚

✚

dx = − cos x +

  • u2du
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SLIDE 40

Odd Powers of cos x and sin x

✟✟✟✟✟ ✯− cos x

  • sin xdx −
  • cos2 x sin xdx = − cos x −
  • cos2 x sin xdx

Now, we solve this integral by substitution: u = cos x, du dx = − sin x − cos x −

  • cos2 x sin xdx = − cos x
  • −
  • u2
  • −du

✚ ✚

dx

  • ✚

✚

dx = − cos x +

  • u2du = − cos x+
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SLIDE 41

Odd Powers of cos x and sin x

✟✟✟✟✟ ✯− cos x

  • sin xdx −
  • cos2 x sin xdx = − cos x −
  • cos2 x sin xdx

Now, we solve this integral by substitution: u = cos x, du dx = − sin x − cos x −

  • cos2 x sin xdx = − cos x
  • −
  • u2
  • −du

✚ ✚

dx

  • ✚

✚

dx = − cos x +

  • u2du = − cos x + u3

3

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SLIDE 42

Odd Powers of cos x and sin x

✟✟✟✟✟ ✯− cos x

  • sin xdx −
  • cos2 x sin xdx = − cos x −
  • cos2 x sin xdx

Now, we solve this integral by substitution: u = cos x, du dx = − sin x − cos x −

  • cos2 x sin xdx = − cos x
  • −
  • u2
  • −du

✚ ✚

dx

  • ✚

✚

dx = − cos x +

  • u2du = − cos x + u3

3 + C

slide-43
SLIDE 43

Odd Powers of cos x and sin x

✟✟✟✟✟ ✯− cos x

  • sin xdx −
  • cos2 x sin xdx = − cos x −
  • cos2 x sin xdx

Now, we solve this integral by substitution: u = cos x, du dx = − sin x − cos x −

  • cos2 x sin xdx = − cos x
  • −
  • u2
  • −du

✚ ✚

dx

  • ✚

✚

dx = − cos x +

  • u2du = − cos x + u3

3 + C

  • sin3 xdx =
slide-44
SLIDE 44

Odd Powers of cos x and sin x

✟✟✟✟✟ ✯− cos x

  • sin xdx −
  • cos2 x sin xdx = − cos x −
  • cos2 x sin xdx

Now, we solve this integral by substitution: u = cos x, du dx = − sin x − cos x −

  • cos2 x sin xdx = − cos x
  • −
  • u2
  • −du

✚ ✚

dx

  • ✚

✚

dx = − cos x +

  • u2du = − cos x + u3

3 + C

  • sin3 xdx = − cos x+
slide-45
SLIDE 45

Odd Powers of cos x and sin x

✟✟✟✟✟ ✯− cos x

  • sin xdx −
  • cos2 x sin xdx = − cos x −
  • cos2 x sin xdx

Now, we solve this integral by substitution: u = cos x, du dx = − sin x − cos x −

  • cos2 x sin xdx = − cos x
  • −
  • u2
  • −du

✚ ✚

dx

  • ✚

✚

dx = − cos x +

  • u2du = − cos x + u3

3 + C

  • sin3 xdx = − cos x + cos3 x

3

slide-46
SLIDE 46

Odd Powers of cos x and sin x

✟✟✟✟✟ ✯− cos x

  • sin xdx −
  • cos2 x sin xdx = − cos x −
  • cos2 x sin xdx

Now, we solve this integral by substitution: u = cos x, du dx = − sin x − cos x −

  • cos2 x sin xdx = − cos x
  • −
  • u2
  • −du

✚ ✚

dx

  • ✚

✚

dx = − cos x +

  • u2du = − cos x + u3

3 + C

  • sin3 xdx = − cos x + cos3 x

3 + C

slide-47
SLIDE 47

Odd Powers of cos x and sin x

✟✟✟✟✟ ✯− cos x

  • sin xdx −
  • cos2 x sin xdx = − cos x −
  • cos2 x sin xdx

Now, we solve this integral by substitution: u = cos x, du dx = − sin x − cos x −

  • cos2 x sin xdx = − cos x
  • −
  • u2
  • −du

✚ ✚

dx

  • ✚

✚

dx = − cos x +

  • u2du = − cos x + u3

3 + C

  • sin3 xdx = − cos x + cos3 x

3 + C

slide-48
SLIDE 48