7.2 1 Section 7.2: Trigonometric Integrals • Objective – Be able to combine integration by substitution with trigonometric identities to integrate trigonometric forms.
7.2 1
Section 7.2: Trigonometric Integrals
• Objective– Be able to combine integration by substitution with
trigonometric identities to integrate trigonometric forms.
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 +=
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∫
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= + + −⎡ ⎤⎣ ⎦
7.2 5
7.2 6
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
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= − + +
7.2 9
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= ∫
7.2 11
7.2 12
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= + =∫
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= + + +
7.2 15
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= − = − = − +∫ ∫ ∫
7.2 17
Example
532
3 5ww w= − +
532 sinsin sin
3 5uu u= − +
( ) ( )53 sin sin2sin sin sin
3 5C
θθ θ= − + +
7.2 18
MORE EXAMPLESFOLLOW; NOT COVERED IN
CLASS
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=
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+ −⎛ ⎞ ⎛ ⎞= ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠∫
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= + − − − +∫ ∫ ∫ ∫
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= + + −⎡ ⎤⎣ ⎦
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