02 trigonometric integrals - handout

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Trig Integrals Trig Integrals Trig Integrals Exercises Techniques of Integration–Trigonometric Integrals Mathematics 54–Elementary Analysis 2 Institute of Mathematics University of the Philippines-Diliman 1 / 26

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Page 1: 02 Trigonometric Integrals - Handout

Trig Integrals Trig Integrals Trig Integrals Exercises

Techniques of Integration–TrigonometricIntegrals

Mathematics 54–Elementary Analysis 2

Institute of MathematicsUniversity of the Philippines-Diliman

1 / 26

Page 2: 02 Trigonometric Integrals - Handout

Trig Integrals Trig Integrals Trig Integrals Exercises∫

sinm x dx or∫

cosm x dx∫

sinm x cosn x dx

Trigonometric IntegralsIntegrals of the form

∫sinm x dx or

∫cosm x dx

Recall∫sinx dx =−cosx+C∫sin2 x dx =

∫1

2(1−cos2x) dx = 1

2x− 1

4sin2x+C

2 / 26

Page 3: 02 Trigonometric Integrals - Handout

Trig Integrals Trig Integrals Trig Integrals Exercises∫

sinm x dx or∫

cosm x dx∫

sinm x cosn x dx

Trigonometric IntegralsIntegrals of the form

∫sinm x dx or

∫cosm x dx

Example.

Consider∫

sin3 x dx.

Note thatsin3 x = sin2 x sinx

= (1−cos2 x) sinx

= sinx−cos2 x sinx

Thus,∫

sin3 x dx =∫ (

sinx− cos 2x sinx)

dx.

Let u = cosx, du =−sinx dx. Therefore,

∫sin3 x dx =−

∫ (1−u2) du =−u+ 1

3u3 +C =−cosx+ 1

3cos3 x+C

3 / 26

Page 4: 02 Trigonometric Integrals - Handout

Trig Integrals Trig Integrals Trig Integrals Exercises∫

sinm x dx or∫

cosm x dx∫

sinm x cosn x dx

Trigonometric IntegralsIntegrals of the form

∫sinm x dx or

∫cosm x dx

∫sinm x dx, m ∈N

m is oddsplit off a factor of sinx

express the rest of the factors in terms of cosx, usingsin2 x = 1−cos2 x

use the substitution u = cosx, du =−sinx dx

m is evenuse the half-angle identity

sin2 x = 1

2(1−cos2x)

4 / 26

Page 5: 02 Trigonometric Integrals - Handout

Trig Integrals Trig Integrals Trig Integrals Exercises∫

sinm x dx or∫

cosm x dx∫

sinm x cosn x dx

Trigonometric IntegralsIntegrals of the form

∫sinm x dx or

∫cosm x dx

∫cosm x dx, m ∈N

m is oddsplit off a factor of cosx

express the rest of the factors in terms of sinx, usingcos2 x = 1− sin2 x

use the substitution u = sinx, du = cosx dx

m is evenuse the half-angle identity

cos2 x = 1

2(1+cos2x)

5 / 26

Page 6: 02 Trigonometric Integrals - Handout

Trig Integrals Trig Integrals Trig Integrals Exercises∫

sinm x dx or∫

cosm x dx∫

sinm x cosn x dx

Example.

Evaluate∫

cos5 x dx

∫cos5 x dx =

∫cos4 x cosx dx

=∫ (

cos2 x)2

cosx dx =∫ (

1− sin2 x)2

cosx dx

=∫ (

1−2sin 2x+ sin4 x)

cosx dx

Let u = sinx, du = cosx dx.∫cos5 x dx =

∫ (1−2u2 +u4) du

= u− 2

3u3 + 1

5u5 +C

= sinx− 2

3sin3 x+ 1

5sin5 x+C

6 / 26

Page 7: 02 Trigonometric Integrals - Handout

Trig Integrals Trig Integrals Trig Integrals Exercises∫

sinm x dx or∫

cosm x dx∫

sinm x cosn x dx

Trigonometric IntegralsIntegrals of the form

∫sinm x cosn x dx

Example.

Evaluate∫

cos3 x sin2 x dx.

∫cos3 x sin2 x dx = ∫

cos2 x sin2 x cosx dx

=∫ (

1− sin2 x)

sin2 x cosx dx

=∫

sin2 x cosx dx−∫

sin4 x cosx dx

let u = sinx du = cosxdx∫cos3 x sin2 x dx =

∫u2 du−

∫u4 du

= 1

3u3 − 1

5u5 +C = 1

3sin3 x− 1

5sin5 x+C

7 / 26

Page 8: 02 Trigonometric Integrals - Handout

Trig Integrals Trig Integrals Trig Integrals Exercises∫

sinm x dx or∫

cosm x dx∫

sinm x cosn x dx

Trigonometric IntegralsIntegrals of the form

∫sinm x cosn x dx

∫sinm x cosn x dx

m is oddsplit off a factor of sinx

express the rest of the factors in terms of cosx, usingsin2 x = 1−cos2 x

use the substitution u = cosx, du =−sinx dx

8 / 26

Page 9: 02 Trigonometric Integrals - Handout

Trig Integrals Trig Integrals Trig Integrals Exercises∫

sinm x dx or∫

cosm x dx∫

sinm x cosn x dx

Trigonometric IntegralsIntegrals of the form

∫sinm x cosn x dx

∫sinm x cosn x dx

n is oddsplit off a factor of cosx

express the rest of the factors in terms of sinx, usingcos2 x = 1− sin2 x

use the substitution u = sinx, du = cosx dx

both m and n are evenuse the half-angle identities

cos2 x = 1

2(1+cos2x) and sin2 x = 1

2(1−cos2x)

use the rule for∫

cosm x dx

9 / 26

Page 10: 02 Trigonometric Integrals - Handout

Trig Integrals Trig Integrals Trig Integrals Exercises∫

sinm x dx or∫

cosm x dx∫

sinm x cosn x dx

Example.

Evaluate∫

sin2 x cos4 x dx.

∫sin2 x cos4 x dx =

∫sin2 x (cos2 x)2dx

=∫ (

1−cos2x

2

)(1+cos2x

2

)2

dx

=∫ (

1−cos2x

2

)(1+cos2x

2

)2

dx

= 1

8

∫ (1+cos2x−cos2 2x−cos3 2x

)dx

= 1

8

∫ [1+cos2x−

(1+cos4x

2

)− (1− sin2 2x)cos2x

]dx

= 1

8

[x+ sin2x

2− 1

2

(x+ sin4x

4

)− 1

2

(sin2x− sin3 2x

3

)]+C

10 / 26

Page 11: 02 Trigonometric Integrals - Handout

Trig Integrals Trig Integrals Trig Integrals Exercises∫

tanm x dx or∫

cotm x dx∫

secn x dx or∫

cscn x dx

Trigonometric IntegralsIntegrals of the form

∫tanm x dx or

∫cotm x dx

Example.

Evaluate∫

tan3 x dx.

∫tanx tan2 x dx =

∫tanx

(sec2 x−1

)dx

=∫

tanx sec2 x dx−∫

tanx dx

let u = tanx, du = sec2 x dx∫tan3 x dx =

∫udu− ln |secx|+C

= 1

2u2 − ln |secx|+C

= 1

2

(tan2 x

)− ln |secx|+C

11 / 26

Page 12: 02 Trigonometric Integrals - Handout

Trig Integrals Trig Integrals Trig Integrals Exercises∫

tanm x dx or∫

cotm x dx∫

secn x dx or∫

cscn x dx

Trigonometric IntegralsIntegrals of the form

∫tanm x dx or

∫cotm x dx

∫tanm x dx

split off a factor of tan2 x and write this as tan2 x = sec2 x−1

use the substitution u = tanx, du = sec2 x dx

∫cotm x dx

split off a factor of cot2 x and write this as cot2 x = csc2 x−1

use the substitution u = cotx, du =−csc2 x dx

12 / 26

Page 13: 02 Trigonometric Integrals - Handout

Trig Integrals Trig Integrals Trig Integrals Exercises∫

tanm x dx or∫

cotm x dx∫

secn x dx or∫

cscn x dx

Example.

Evaluate∫

cot4 3x dx.

∫cot2 3x cot2 3x dx =

∫cot2 3x

(csc2 3x−1

)dx

=∫ (

cot2 3x csc2 3x−cot2 3x)

dx

=∫ (

cot2 3x csc2 3x−csc2 3x+1)

dx

=∫ (

cot2 3x csc2 3x)

dx+ 1

3cot3x+x+C

let u = cot3x, du =−3csc2 3x dx∫cot4 3x dx = −1

3

∫u2 du+ 1

3cot3x+x+C

= −1

9u3 + 1

3cot3x+x+C

= −1

9cot3 3x+ 1

3cot3x+x+C

13 / 26

Page 14: 02 Trigonometric Integrals - Handout

Trig Integrals Trig Integrals Trig Integrals Exercises∫

tanm x dx or∫

cotm x dx∫

secn x dx or∫

cscn x dx

Trigonometric IntegralsIntegrals of the form

∫secn x dx or

∫cscn x dx

Example.

Evaluate∫

csc6 x dx.

∫csc6 x dx =

∫(csc2 x)2 csc2 x dxdx

=∫ (

1+cot2 x)

csc2 xdx

=∫

(1+2cot2 x+cot4 x)csc2 x dx

let u = cotx ⇒ du =−csc2 x dx∫csc6 x dx =−∫

(1+2u2 +u4) du

=−(cotx+ 2cot3 x

3+ cot5 x

5

)+C

14 / 26

Page 15: 02 Trigonometric Integrals - Handout

Trig Integrals Trig Integrals Trig Integrals Exercises∫

tanm x dx or∫

cotm x dx∫

secn x dx or∫

cscn x dx

Trigonometric IntegralsIntegrals of the form

∫secn x dx or

∫cscn x dx

∫secn xdx

n is evensplit off a factor of sec2 x.express the rest of the factors in terms of tanx, usingsec2 x = 1+ tan2 xuse the substitution u = tanx, du = sec2 xdx.

∫cscn xdx

n is evensplit off a factor of csc2 x.express the rest of the factors in terms of cotx, usingcsc2 x = 1+cot2 xuse the substitution u = cotx, du =−csc2 xdx

15 / 26

Page 16: 02 Trigonometric Integrals - Handout

Trig Integrals Trig Integrals Trig Integrals Exercises∫

tanm x dx or∫

cotm x dx∫

secn x dx or∫

cscn x dx

Example.

Evaluate∫

sec3 x dx.

Note that sec3 x = secx sec2 x. By IBP,

u = secx , dv = sec2 x dxdu = secx tanx dx , v = tanx dx

∫sec3 x dx = secx tanx−

∫tanx(secx tanx) dx

= secx tanx−∫

tan2 x secx dx

= secx tanx−∫

(sec2 x−1)secx dx∫sec3 x dx = secx tanx−

∫sec3 x dx+

∫secx dx

2∫

sec3 xdx = secx tanx+ ln |secx+ tanx|+C

∴∫

sec3 xdx = 1

2(secx tanx+ ln |secx+ tanx|)+C

16 / 26

Page 17: 02 Trigonometric Integrals - Handout

Trig Integrals Trig Integrals Trig Integrals Exercises∫

tanm x dx or∫

cotm x dx∫

secn x dx or∫

cscn x dx

Trigonometric IntegralsIntegrals of the form

∫secn x dx or

∫cscn x dx

∫secn xdx

n is oddsplit off a factor of sec2 xuse IBP with dv = sec2 x dx and u to be the remaining factors

∫cscn xdx

n is oddsplit off a factor of csc2 xuse IBP, with dv = csc2 x dx and u to be the remaining factors

17 / 26

Page 18: 02 Trigonometric Integrals - Handout

Trig Integrals Trig Integrals Trig Integrals Exercises∫

tanm x secn x dx or∫

cotm x cscn x dx∫

sin mx cos nx dx,∫

sin mx sin nx dx or∫

cos mx cos nx dx

Trigonometric IntegralsIntegrals of the form

∫tanm x secn x dx or

∫cotm x cscn x dx

Example.

Evaluate∫

tan3 x sec2 x dx.

∫tan3 x sec2 x dx =

∫tan2 x secx secx tanx dx

=∫ (

sec2 x −1)

secx secx tanx dx

=∫ (

sec3 x− secx)

secx tanx dx

let u = secx, du = secx tanx dx∫tan3 x sec2 x dx =

∫ (u3 −u

)du

= 1

4sec4 x− 1

2sec2 x+C

18 / 26

Page 19: 02 Trigonometric Integrals - Handout

Trig Integrals Trig Integrals Trig Integrals Exercises∫

tanm x secn x dx or∫

cotm x cscn x dx∫

sin mx cos nx dx,∫

sin mx sin nx dx or∫

cos mx cos nx dx

Trigonometric IntegralsIntegrals of the form

∫tanm x secn x dx or

∫cotm x cscn x dx

∫tanm x secn x dx

m is oddsplit off a factor of secx tanxexpress the rest of the factors in terms of secx using the identitytan2 x = sec2 x−1use the substitution u = secx, du = secx tanx dx

∫cotm x cscn x dx

m is oddsplit off a factor of cscx cotxexpress the rest of the factors in terms of cscx using the identitycot2 x = csc2 x−1use the substitution u = cscx, du =−cscx cotx dx

19 / 26

Page 20: 02 Trigonometric Integrals - Handout

Trig Integrals Trig Integrals Trig Integrals Exercises∫

tanm x secn x dx or∫

cotm x cscn x dx∫

sin mx cos nx dx,∫

sin mx sin nx dx or∫

cos mx cos nx dx

Trigonometric IntegralsIntegrals of the form

∫tanm x secn x dx or

∫cotm x cscn x dx

∫tanm x secn x dx

n is evensplit off a factor of sec2 xexpress the rest of the factors in terms of tanx using the identitysec2 x = 1+ tan2 xuse the substitution u = tanx, du = sec2 x dx

∫cotm x cscn x dx

n is evensplit off a factor of csc2 xexpress the rest of the factors in terms of cotx using the identitycsc2 x = 1+cot2 xuse the substitution u = cotx, du =−csc2 x dx

20 / 26

Page 21: 02 Trigonometric Integrals - Handout

Trig Integrals Trig Integrals Trig Integrals Exercises∫

tanm x secn x dx or∫

cotm x cscn x dx∫

sin mx cos nx dx,∫

sin mx sin nx dx or∫

cos mx cos nx dx

Example.

Evaluate∫

cot2 x cscx dx.

∫cot2 x cscx dx =

∫(csc2 x−1)cscx dx

=∫

(csc3 x−cscx) dx

=∫

csc3 x dx− ln |cscx−cotx|

Exercise:∫

csc3 x dx =−1

2cscx cotx+ 1

2ln |cscx−cotx|+C

=−1

2cscx cotx− 1

2ln |cscx−cotx|+C

21 / 26

Page 22: 02 Trigonometric Integrals - Handout

Trig Integrals Trig Integrals Trig Integrals Exercises∫

tanm x secn x dx or∫

cotm x cscn x dx∫

sin mx cos nx dx,∫

sin mx sin nx dx or∫

cos mx cos nx dx

Example.

Evaluate∫ p

tanx sec4 x dx.

∫ ptanx sec4 x dx =

∫ ptanx sec2 x sec2 x dx

=∫ p

tanx(1+ tan2 x)sec2 x dx

=∫ (p

tanx+√

tan5 x)

sec2 x dx

= 2

3

√tan3 x+ 2

7

√tan7 x+C

22 / 26

Page 23: 02 Trigonometric Integrals - Handout

Trig Integrals Trig Integrals Trig Integrals Exercises∫

tanm x secn x dx or∫

cotm x cscn x dx∫

sin mx cos nx dx,∫

sin mx sin nx dx or∫

cos mx cos nx dx

Trigonometric IntegralsIntegrals of the form

∫tanm x secn x dx or

∫cotm x cscn x dx

∫tanm x secn x dx

m is even and n is oddexpress the even power of tanx in terms of secx using theidentity tan2 x = sec2 x−1

use the rule for∫

secm x dx

∫cotm x cscn x dx

m is even and n is oddexpress the even power of cotx in terms of cscx using theidentity cot2 x = csc2 x−1

use the rule for∫

cscm x dx

23 / 26

Page 24: 02 Trigonometric Integrals - Handout

Trig Integrals Trig Integrals Trig Integrals Exercises∫

tanm x secn x dx or∫

cotm x cscn x dx∫

sin mx cos nx dx,∫

sin mx sin nx dx or∫

cos mx cos nx dx

Trigonometric IntegralsF. Integrals of the form

∫sin mx cos nxdx,

∫sin mx sin nxdx or

∫cos mx cos nxdx

Recall. Product to Sum Formula

sinmx cosnx = 1

2[sin(m+n)x+ sin(m−n)x],

sinmx sinnx = −1

2[cos(m+n)x−cos(m−n)x],

cosmx cosnx = 1

2[cos(m+n)x+cos(m−n)x].

24 / 26

Page 25: 02 Trigonometric Integrals - Handout

Trig Integrals Trig Integrals Trig Integrals Exercises∫

tanm x secn x dx or∫

cotm x cscn x dx∫

sin mx cos nx dx,∫

sin mx sin nx dx or∫

cos mx cos nx dx

Example.

Evaluate∫

cos3x cos5x dx.

∫cos3x cos5x dx = 1

2

∫(cos(3x+5x)+cos(3x−5x)) dx

= 1

2

∫(cos8x+cos2x) dx

= 1

2

(1

8sin8x+ 1

2sin2x

)+C

= 1

16sin8x+ 1

4sin2x+C

25 / 26

Page 26: 02 Trigonometric Integrals - Handout

Trig Integrals Trig Integrals Trig Integrals Exercises

Exercises

Evaluate the following integrals.

1

∫ 1

0sin2πx cos2πx dx

2

∫cos3 xp

sinxdx

3

∫csc4 x

cot2 xdx

4

∫cos 4x cos 3x dx

5

∫tan3(lnx)sec8(lnx)

xdx

26 / 26