7.2 Trigonometric IntegralsIn this section, we introduce techniques for evaluating integrals of the form

dxxx

dxxx

dxxx

nm

nm

nm

csccot

sectan

cossin

where either m or n is a nonnegative integer

and occasionally, integrals involving the product of sines and cosines of two different angles.

REFER to the Handout for Trigonometric Integrals

Integrals Involving Sine and Cosine

dxxx nm )(cos)(sinA.

1. If the power of the sine is odd and positive, save one sine factor and convert the remaining factors to cosine. Then, expand and integrate.

xdxxx

xdxxxxdxx

nk

nknk

sincoscos1

sincossincossin

2

212

odd convert to cosine save for du

2. If the power of the cosine is odd and positive, save one cosine factor and convert the remaining factors to sine. Then, expand and integrate.

xdxxx

xdxxxxdxx

km

kmkm

sinsin1sin

coscossincossin

2

212

oddconvert to sine save for du

3. If the powers of both the sine and cosine are even and nonnegative, make repeated use of the identities

to convert the integrand to odd powers of the cosine. Then, proceed as in case 2.

It is sometimes helpful to use the identity

2

2cos1sin 2 x

x

and 2

2cos1cos 2 x

x

xxx 2sin2

1cossin

Examples: Integrals Involving Sine and Cosine

Evaluate the integral.

1)

2)

Cxx

xdxxxdxx

xdxxx

dxxxxxdxx

7

cos

5

cos

sincossincos

sincoscos1

sincossincossin

75

64

42

4243

960

203

32

1

10

1

8

1

3

1

2

1

2

1

5

)2(sin

2

1

3

)2(sin)2sin(

2

1

)2cos()2(sin)2cos()2(sin2)2cos(

)2cos()2(sin)2(sin21

2cos)2(sin1

2cos2cos2cos

12/

0

53

12/

0

42

12/

0

42

212/

0

2

12/

0

12/

0

45

xxx

dxxxxxx

dxxxx

dxxx

dxxxdxx

3)

4)

Cxxx

xdxxdxdx

dxxx

dxxx

dxx

xdx

4sin32

12sin

4

1

8

3

4cos8

12cos

2

1

8

3

2

4cos1

4

1

2

2cos

4

1

4

2cos

2

2cos

4

1

2

2cos1cos

224

1628

14sin

4

1

8

1

4cos18

1

2

4cos1

4

1

2sin4

12sin

2

1cossin

2/

0

2/

0

2/

0

2/

0

222/

0

2/

0

22

xx

dxxdxx

dxxdxxxdxx

Integrals Involving Secants and Tangents

dxxx nm )(sec)(tanB.

1. If the power of the tangent is odd and positive, save a secant-tangent factor and convert the remaining factors to secants. Then, expand and integrate.

xdxxxx

xdxxxxxdxx

nk

nknk

tansecsec1sec

tansecsectansectan

12

1212

odd convert to secants save for du

2. If the power of the secant is even and positive, save a secant squared factor and convert the remaining factors to tangents. Then, expand and integrate.

xdxxx

xdxxxxdxx

km

kmkm

212

2122

sectan1tan

secsectansectan

even convert to tangents save for du

3. If there are no secant factors and the power of the tangent is even and positive, convert a tangent squared factor to secants; then expand and repeat if necessary.

xdxdxxx

dxxxdxxxxdx

mm

mmm

222

2222

tansectan

1sectantantantan

convert to secants

4. If the integral is of the form , where n is odd and positive, possibly use integration by parts.

5. If none of the first four cases apply, try converting to sines and cosines.

xdxnsec

Examples:Integrals Involving Secants and Tangents

Evalute the integral.

1)

Cxx

dxxxxx

dxxxxx

dxxxxx

xdxxdxx

x

2/12/3

2/32/1

22/3

22/3

32/13

sec2sec3

2

tansecsecsec

tansec1secsec

tansectansec

tansecsec

tan

2)

Cxx

dxxxx

dxxxx

dxxxxxdxx

6

3tan

4

3tan

3

1

3sec33tan3tan3

1

3sec3tan3tan1

3sec3tan3sec3tan3sec

64

253

222

23234

Power Rule

Power Rule

3)

3

2

4tan

3

tan

1secsectan

tansectan

1sectantantantan

4/

0

3

4/

0

24/

0

22

4/

0

24/

0

22

4/

0

4/

0

22224/

0

4

xxx

dxxxdxx

xdxxdxx

dxxxdxxxxdx

4) Since the first two cases do not apply, try converting to sines and cosines.

CxCx

dxxxdxx

x

xdxx

x

cscsin

cossinsin

cos

cos

1

tan

sec

1

22

2

Integrals Involving Cotangent and Cosecant

NOTE: The guidelines for integrals involving cotangent and cosecant would be similar to that of integrals involving tangent and secant. Refer to your handout.

Example:

Cxx

dxxxx

dxxxx

dxxxxdxxx

7

cot

5

cot

csccotcot

csccotcot1

csccotcsccotcsc

75

264

242

24244

Integrals Involving Sine-Cosine Products with Different angles

In such instances, we use the following product-to-sum identities.

xnmxnmnxmx

xnmxnmnxmx

xnmxnmnxmx

)sin()sin(2

1cossin

)cos()cos(2

1sinsin

)cos()cos(2

1coscos

Example:

Cxx

Cxx

dxxxxdxx

2

cos

18

9cos

cos9

9cos

2

1

sin9sin2

14cos5sin