02 trigonometric integrals

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

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7/27/2019 02 Trigonometric Integrals

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Trig Integrals Trig Integrals Trig Integrals Exercises

Techniques of Integration–TrigonometricIntegrals

Mathematics 54–Elementary Analysis 2

Institute of Mathematics

University of the Philippines-Diliman

1/26

7/27/2019 02 Trigonometric Integrals

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7/27/2019 02 Trigonometric Integrals

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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

=sin2

x  sin x 

= (1−cos2 x ) sin x 

= sin x −cos2x sin x 

Thus,

sin3

x dx =

sin x − cos 2

x sin x 

dx .

Let u = cos x , du =−sin x dx . Therefore,

sin3

x dx =−

1−u 

2

du =−u + 1

3u 

3+C =−cos x + 1

3cos3

x +C 

3/26

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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 odd

split off a factor of sin x 

express the rest of the factors in terms of cos x , using sin2

x = 1−cos2x 

use the substitution u = cos x , du =−sin x dx 

m is even

use the half-angle identity 

sin2x = 1

2(1−cos2x )

4/26

m

m

m n

7/27/2019 02 Trigonometric Integrals

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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 odd

split off a factor of cos x 

express the rest of the factors in terms of sin x , using cos2

x = 1−sin2x 

use the substitution u = sin x , du = cos x dx 

m is even

use the half-angle identity 

cos2x = 1

2(1+cos2x )

5/26

T i I l T i I l T i I l E i

i m d

m d

i m n d

7/27/2019 02 Trigonometric Integrals

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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  =

cos4x cos x dx 

= cos2x 

2cos x dx 

=1

−sin2

x 2

cos x dx 

=

1−2sin 2x +sin4

cos x dx 

Let u = sin x , du = cos x dx .

cos5 x dx  =

1−2u 2+u 4

du 

= u − 2

3u 

3+ 1

5u 

5+C 

= sin x − 2

3sin3

x + 1

5sin5

x +C 

6/26

T i I t l T i I t l T i I t l E i

i m d

m d

i m n d

7/27/2019 02 Trigonometric Integrals

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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 sin2x dx .

cos

3

x sin2

x dx  =cos2 x sin2 x cos x dx 

=

1−sin2x 

sin2x cos x dx 

=

sin2x cos x dx −

sin4

x cos x dx 

let u = sin x du = cos xdx cos3

x sin2x dx  =

2du −

4du 

=1

3

u 3

−1

5

u 5

+C 

=1

3

sin3x 

−1

5

sin5x 

+C 

7/26

Trig Integrals Trig Integrals Trig Integrals Exercises

sinm x dx or

cosm x dx

sinm x cosn x dx

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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 sin x 

express the rest of the factors in terms of cos x , using 

sin2x = 1−cos2

use the substitution u = cos x , du =−sin x dx 

8/26

Trig Integrals Trig Integrals Trig Integrals Exercises

sinm x dx or

cosm x dx

sinm x cosn x dx

7/27/2019 02 Trigonometric Integrals

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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 odd

split off a factor of cos x 

express the rest of the factors in terms of sin x , using cos2

x = 1−sin2x 

use the substitution u = sin x , du = cos x dx 

both m and n are even

use the half-angle identities

cos2x = 1

2(1+cos2x ) and sin2

x = 1

2(1−cos2x )

use the rule for

cosm 

x dx 

9/26

Trig Integrals Trig Integrals Trig Integrals Exercises

sinm x dx or

cosm x dx

sinm x cosn x dx

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Trig Integrals Trig Integrals Trig Integrals Exercises

sin x dx or

cos x dx 

sin x  cos x dx 

Example.

Evaluate sin2x cos4

x dx .

sin2

x cos4x dx =

sin2

x (cos2x )2

dx 

=1

−cos2x 

21

+cos2x 

2

2

dx 

=

1−cos2x 

2

1+2cos2x +cos2 2x 

4

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

Trig Integrals Trig Integrals Trig Integrals Exercises

tanm x dx or cotm x dx

secn x dx or

cscn x dx

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Trig Integrals Trig Integrals Trig Integrals Exercises

tan x dx or cot x dx 

sec x dx or

csc x dx 

Trigonometric IntegralsIntegrals of the form

tanm 

x dx  or

cotm x dx 

Example.

Evaluate

tan3

x dx .

tan x tan

2

x dx  = tan x 

sec

2

x −1

dx 

=

tan x sec2x dx −

tan x dx 

let u = tan x , du = sec2x dx 

tan3 x dx  =

u du − ln |sec x |+C 

= 1

2u 

2− ln |sec x |+C 

=

1

2tan2

x −ln

|sec x 

|+C 

11/26

Trig Integrals Trig Integrals Trig Integrals Exercises

tanm x dx or cotm x dx 

secn x dx or

cscn x dx 

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g g g g g g

Trigonometric IntegralsIntegrals of the form

tanm 

x dx  or

cotm x dx 

tanm 

x dx 

split off a factor of tan2x  and write this as tan2

x = sec2x −1

use the substitution u 

=tan x , du 

=sec2

x dx 

cotm x dx 

split off a factor of cot2x  and write this as cot2

x = csc2x −1

use the substitution u = cot x , du =−csc2x dx 

12/26

Trig Integrals Trig Integrals Trig Integrals Exercises

tanm x dx or cotm x dx 

secn x dx or

cscn x dx 

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g g g g g g

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

2du + 1

3cot3x +x +C 

= −1

9u 

3+ 1

3cot3x +x +C 

= −1

9 cot

3

3x +

1

3 cot3x +

x +

13/26

Trig Integrals Trig Integrals Trig Integrals Exercises

tanm x dx or cotm x dx 

secn x dx or

cscn x dx 

7/27/2019 02 Trigonometric Integrals

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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+cot2x 

csc2xdx 

=

(1+2cot2x +cot4

x )csc2x dx 

let u = cot x ⇒ du =−csc2 x dx csc6

x dx  =−

(1+2u 2+u 

4) du 

=−

cot x + 2cot3x 

3+ cot5

5

+C 

14/26

Trig Integrals Trig Integrals Trig Integrals Exercises

tanm x dx or cotm x dx 

secn x dx or

cscn x dx 

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Trigonometric IntegralsIntegrals of the form

secn 

x dx  or

cscn x dx 

secn xdx 

n is even

split off a factor of sec2x .

express the rest of the factors in terms of tan x , using 

sec2

x = 1+ tan2

x use the substitution u = tan x , du = sec2

xdx .

cscn 

xdx 

n is evensplit off a factor of csc2

x .

express the rest of the factors in terms of cot x , using 

csc2x = 1+cot2

use the substitution u 

=cot x , du 

=−csc2

xdx 

15/26

Trig Integrals Trig Integrals Trig Integrals Exercises

tanm x dx or cotm x dx 

secn x dx or

cscn x dx 

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Example.

Evaluate

sec3

x dx .

Note that sec3x = sec x sec2

x . By IBP,

u = sec x  , dv = sec2x dx 

du = sec x tan x dx  , v = tan x dx 

sec3

x dx  = sec x tan x −

tan x (sec x tan x ) dx 

= sec x tan x −

tan2x sec x dx 

=sec x tan x 

−(sec2x 

−1)sec x dx 

sec3x dx  = sec x tan x −

sec3

x dx +

sec x dx 

2

sec3

xdx  = sec x tan x + ln |sec x + tan x |+C 

sec3

xdx  = 1

2(sec x tan x + ln |sec x + tan x |)+C 

16/26

Trig Integrals Trig Integrals Trig Integrals Exercises

tanm x dx or cotm x dx 

secn x dx or

cscn x dx 

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Trigonometric IntegralsIntegrals of the form

secn 

x dx  or

cscn x dx 

secn 

xdx 

n is odd

split off a factor of sec2x 

use IBP with dv = sec2 x dx  and u to be the remaining factors

cscn 

xdx 

n is oddsplit off a factor of csc2

use IBP, with dv = csc2x dx  and u to be the remaining factors

17/26

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 

7/27/2019 02 Trigonometric Integrals

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Trigonometric IntegralsIntegrals of the form

tanm 

x secn x dx  or

cotm 

x cscn x dx 

Example.

Evaluate

tan3

x sec2x dx .

tan

3

x sec2

x dx  = tan

2

x sec x sec x tan x dx 

=

sec2x −1

sec x sec x tan x dx 

=

sec3x −sec x sec x tan x dx 

let u = sec x , du = sec x tan x dx tan3

x sec2x dx  =

3−u 

du 

=

1

4

sec4x 

1

2

sec2x 

+C 

18/26

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 

7/27/2019 02 Trigonometric Integrals

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Trigonometric IntegralsIntegrals of the form

tanm 

x secn x dx  or

cotm 

x cscn x dx 

tanm x secn x dx 

m is odd

split off a factor of sec x tan x 

express the rest of the factors in terms of sec x using the identity 

tan2

x = sec2

x −1use the substitution u = sec x , du = sec x tan x dx 

cotm 

x cscn x dx 

m is oddsplit off a factor of csc x cot x 

express the rest of the factors in terms of csc x  using the identity 

cot2x = csc2

x −1

use the substitution u = csc x , du =−csc x cot x dx 

19/26

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 

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Trigonometric IntegralsIntegrals of the form

tanm 

x secn x dx  or

cotm 

x cscn x dx 

tanm x secn x dx 

n is even

split off a factor of sec2x 

express the rest of the factors in terms of tan x using the identity 

sec2

x = 1+ tan2

x use the substitution u = tan x , du = sec2

x dx 

cotm 

x cscn x dx 

n is evensplit off a factor of csc2

express the rest of the factors in terms of cot x using the identity 

csc2x = 1+cot2

use the substitution u = cot x , du =−csc2x dx 

20/26

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 

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Example.

Evaluate cot2x csc x dx .

cot2

x csc x dx  =

(csc2x −1)csc x dx 

=(csc3 x −csc x ) dx 

=

csc3x dx − ln |csc x −cot x |

Exercise:

csc3

x dx =−1

2 csc x cot x +1

2 ln |csc x −cot x |+C 

=−1

2csc x cot x − 1

2ln |csc x −cot x |+C 

21/26

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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 

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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 odd

express the even power of tan x  in terms of sec x using the

identity tan2x = 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 cot x in terms of csc x using the

identity cot2x = csc2

x −1

use the rule for

cscm 

x dx 

23/26

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 

i i l

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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 +

)x +sin(

m −

)x 

],

sin mx sin nx  = −1

2[cos(m +n )x −cos(m −n )x ],

cos mx cos nx  = 1

2[cos(m +n )x +cos(m −n )x ].

24/26

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 

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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

Trig Integrals Trig Integrals Trig Integrals Exercises

E i

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Exercises

Evaluate the following integrals.

1

1

0sin2

πx cos2πx dx 

2 cos3

 sin x  dx 

3

csc4

cot2 x dx 

4 cos 4x  cos 3x dx 

5

tan3(ln x )sec8(ln x )

x dx 

26/26