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7.2 1 Section 7.2: Trigonometric Integrals • Objective Be able to combine integration by substitution with trigonometric identities to integrate trigonometric forms.

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Page 1: Section 7.2: Trigonometric Integrals 7.2.pdf7.2 1 Section 7.2: Trigonometric Integrals •Objective – Be able to combine integration by substitution with trigonometric identities

7.2 1

Section 7.2: Trigonometric Integrals

• Objective– Be able to combine integration by substitution with

trigonometric identities to integrate trigonometric forms.

Page 2: Section 7.2: Trigonometric Integrals 7.2.pdf7.2 1 Section 7.2: Trigonometric Integrals •Objective – Be able to combine integration by substitution with trigonometric identities

2

Useful Identities

Pythagorean Identities

Half-Angle Identities

2 2sin cos 1x x+ =2 21 tan secx x+ =2 21 cot cscx x+ =

2 1 cos 2sin2

xx −=

2 1 cos 2cos2

xx +=

Page 3: Section 7.2: Trigonometric Integrals 7.2.pdf7.2 1 Section 7.2: Trigonometric Integrals •Objective – Be able to combine integration by substitution with trigonometric identities

7.2 3

Common Trigonometric Forms

1. sin and cosn nx dx x dx∫ ∫

2. sin cosm nx xdx∫

4. sin cos , sin sin , cos cosmx nxdx mx nxdx mx nxdx∫ ∫ ∫

3. tan secm nx xdx∫

Page 4: Section 7.2: Trigonometric Integrals 7.2.pdf7.2 1 Section 7.2: Trigonometric Integrals •Objective – Be able to combine integration by substitution with trigonometric identities

7.2 4

Product Trig Identities

( ) ( )11. sin cos sin sin2

mx nxdx m n x m n x= + + −⎡ ⎤⎣ ⎦

( ) ( )12. sin sin cos cos2

mx nx m n x m n x= − + − −⎡ ⎤⎣ ⎦

( ) ( )13. cos cos cos cos2

mx nx m n x m n x= + + −⎡ ⎤⎣ ⎦

Page 5: Section 7.2: Trigonometric Integrals 7.2.pdf7.2 1 Section 7.2: Trigonometric Integrals •Objective – Be able to combine integration by substitution with trigonometric identities

7.2 5

Page 6: Section 7.2: Trigonometric Integrals 7.2.pdf7.2 1 Section 7.2: Trigonometric Integrals •Objective – Be able to combine integration by substitution with trigonometric identities

7.2 6

Page 7: Section 7.2: Trigonometric Integrals 7.2.pdf7.2 1 Section 7.2: Trigonometric Integrals •Objective – Be able to combine integration by substitution with trigonometric identities

7.2 7

Typed Example (n even)

Solution:

4Solve sin 6x dx∫

6 , 6u x du dx= =

4 41sin 6 sin6

x dx u du=∫ ∫

( )21 1 2cos 2 cos 224

u u du= − +∫

21 1 cos 26 2

u du−⎛ ⎞= ⎜ ⎟⎝ ⎠∫

221 sin6

u du⎡ ⎤= ⎣ ⎦∫

Half-Angle Formula

Page 8: Section 7.2: Trigonometric Integrals 7.2.pdf7.2 1 Section 7.2: Trigonometric Integrals •Objective – Be able to combine integration by substitution with trigonometric identities

8

Example 2 Continued – Type 1 (n even)

( )1 1 12cos 2 1 cos 424 24 48

du udu u du= − + +∫ ∫ ∫

3 1 12cos 2 4cos 448 24 192

du udu udu= − +∫ ∫ ∫

( )3 1 16 sin12 sin 2448 24 192

x x x C= − + +

3 1 1sin12 sin 248 24 192

x x x C= − + +

Page 9: Section 7.2: Trigonometric Integrals 7.2.pdf7.2 1 Section 7.2: Trigonometric Integrals •Objective – Be able to combine integration by substitution with trigonometric identities

7.2 9

Page 10: Section 7.2: Trigonometric Integrals 7.2.pdf7.2 1 Section 7.2: Trigonometric Integrals •Objective – Be able to combine integration by substitution with trigonometric identities

10

Typed Example

2 2sin cosFind x x dx∫

Using the double angle formula for 2 2sin cosx and x

( ) ( )2 2 1 1sin cos 1 cos2 1 cos22 2

x x dx x x dx= − +∫ ∫

( ) ( )1 1 cos2 1 cos24

x x dx= − +∫

21 1 cos 24

x dx= −∫

( )1 1 cos48

x dx= −∫1 1 sin 48 4

x x C⎡ ⎤= − +⎢ ⎥⎣ ⎦

( )1 1 1 cos44 2

x dx= −∫21 sin 24

x dx= ∫

Page 11: Section 7.2: Trigonometric Integrals 7.2.pdf7.2 1 Section 7.2: Trigonometric Integrals •Objective – Be able to combine integration by substitution with trigonometric identities

7.2 11

Page 12: Section 7.2: Trigonometric Integrals 7.2.pdf7.2 1 Section 7.2: Trigonometric Integrals •Objective – Be able to combine integration by substitution with trigonometric identities

7.2 12

Page 13: Section 7.2: Trigonometric Integrals 7.2.pdf7.2 1 Section 7.2: Trigonometric Integrals •Objective – Be able to combine integration by substitution with trigonometric identities

7.2 13

Typed Example

5 6tan secFind x x dx∫5 6tan secx x dx∫ 5 4 2tan sec secx x x dx= ∫

5 4 2tan sec secx x x dx= ∫ ( )25 2 2tan sec secx x x dx= ∫

( )25 2 2tan 1 tan secx x x dx= +∫ 2 2(sec 1 tan )because x x= +

( )25 21 , tanu u du where u x= + =∫

Page 14: Section 7.2: Trigonometric Integrals 7.2.pdf7.2 1 Section 7.2: Trigonometric Integrals •Objective – Be able to combine integration by substitution with trigonometric identities

7.2 14

Continued

( )25 21 , tanu u du where u x= + =∫

( )5 2 41 2u u u du= + +∫ ( )5 7 92u u u du= + +∫6 10

826 8 10u uu C= + + +

6 1081

6 4 10u uu C= + + +

6 8 10tan tan tan6 4 10

x x x C= + + +

Page 15: Section 7.2: Trigonometric Integrals 7.2.pdf7.2 1 Section 7.2: Trigonometric Integrals •Objective – Be able to combine integration by substitution with trigonometric identities

7.2 15

Page 16: Section 7.2: Trigonometric Integrals 7.2.pdf7.2 1 Section 7.2: Trigonometric Integrals •Objective – Be able to combine integration by substitution with trigonometric identities

7.2 16

Example

( )5cos cos sin dθ θ θ∫Evaluate the integral

sin , cosu du dθ θ θ= =

( )5cos cos sin dθ θ θ =∫ 5cos u du∫4cos cosu u du= ∫ ( )221 sin cosu u du= −∫

sin , cosLet w u dw u du= =

( ) ( ) ( )2 22 2 2 41 sin cos 1 1 2u u du w dw w w dw= − = − = − +∫ ∫ ∫

Page 17: Section 7.2: Trigonometric Integrals 7.2.pdf7.2 1 Section 7.2: Trigonometric Integrals •Objective – Be able to combine integration by substitution with trigonometric identities

7.2 17

Example

532

3 5ww w= − +

532 sinsin sin

3 5uu u= − +

( ) ( )53 sin sin2sin sin sin

3 5C

θθ θ= − + +

Page 18: Section 7.2: Trigonometric Integrals 7.2.pdf7.2 1 Section 7.2: Trigonometric Integrals •Objective – Be able to combine integration by substitution with trigonometric identities

7.2 18

MORE EXAMPLESFOLLOW; NOT COVERED IN

CLASS

Page 19: Section 7.2: Trigonometric Integrals 7.2.pdf7.2 1 Section 7.2: Trigonometric Integrals •Objective – Be able to combine integration by substitution with trigonometric identities

7.2 19

Example 3 – Type 2 (m or n odd)

Solution:

( )3Solve sin 2 cos2t t dt∫

( )( )12 21 cos 2 cos2 sin 2t t t dt= −∫

( ) ( )3 7

2 21 1cos 2 cos 23 7

t t C= − + +

( ) ( ) ( )13 2 2sin 2 cos2 sin 2 sin 2 cos2t t dt t t t dt=∫ ∫

( )1

2 2 11 , cos22

u u du u t⎛ ⎞= − − =⎜ ⎟⎝ ⎠∫

1 52 21

2u u du⎡ ⎤

= − −⎢ ⎥⎣ ⎦∫

cos2because u t=

Page 20: Section 7.2: Trigonometric Integrals 7.2.pdf7.2 1 Section 7.2: Trigonometric Integrals •Objective – Be able to combine integration by substitution with trigonometric identities

7.2 20

Example 4 – Type 2 (m and n even)6 2Solve cos sin dθ θ θ∫

6 2cos sin dθ θ θ∫

( )3 41 1 2cos 2 2cos 2 cos 216

dθ θ θ θ= + − −∫

( ) ( )221 1 1 12cos2 1 sin 2 cos2 1 cos4 16 16 8 64

d d d dθ θ θ θ θ θ θ θ= + − − − +∫ ∫ ∫ ∫

( )32 1 cos2cos2

dθθ θ−⎛ ⎞= ⎜ ⎟⎝ ⎠∫

31 cos2 1 cos22 2

dθ θ θ+ −⎛ ⎞ ⎛ ⎞= ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠∫

Page 21: Section 7.2: Trigonometric Integrals 7.2.pdf7.2 1 Section 7.2: Trigonometric Integrals •Objective – Be able to combine integration by substitution with trigonometric identities

7.2 21

Example 4 Continued – Type 2 (m and n even)

( )21 1 1 1 1sin 2 2cos2 4cos4 1 cos816 16 64 128 128

d d d dθ θ θ θ θ θ θ θ= + ⋅ − − − +∫ ∫ ∫ ∫ ∫

31 1 1 1 1 1sin 2 sin 4 sin816 48 64 128 128 1024

Cθ θ θ θ θ θ= + − − − − +

35 1 1 1sin 2 sin 4 sin 8128 48 128 1024

Cθ θ θ θ= + − − +

( ) ( )221 1 1 12cos2 1 sin 2 cos2 1 cos4 16 16 8 64

d d d dθ θ θ θ θ θ θ θ= + − − − +∫ ∫ ∫ ∫

Page 22: Section 7.2: Trigonometric Integrals 7.2.pdf7.2 1 Section 7.2: Trigonometric Integrals •Objective – Be able to combine integration by substitution with trigonometric identities

7.2 22

Type 3 Product Identities

( ) ( )11. sin cos sin sin2

mx nxdx m n x m n x= + + −⎡ ⎤⎣ ⎦

( ) ( )12. sin sin cos cos2

mx nx m n x m n x= − + − −⎡ ⎤⎣ ⎦

( ) ( )13. cos cos cos cos2

mx nx m n x m n x= + + −⎡ ⎤⎣ ⎦

Page 23: Section 7.2: Trigonometric Integrals 7.2.pdf7.2 1 Section 7.2: Trigonometric Integrals •Objective – Be able to combine integration by substitution with trigonometric identities

7.2 23

Example 5 – Type 3

Solution:

Solve cos cos 4y ydy∫

( )1cos cos 4 cos5 cos 32

y ydy y y dy= + −⎡ ⎤⎣ ⎦∫ ∫

( )1 1sin 5 sin 310 6

y y C= − − +

1 1sin 5 sin 310 6

y y C= + +

( ) ( )1 cos cos cos cos2

mx nx m n x m n x= + + −⎡ ⎤⎣ ⎦

m=1, n=4