Math 104-006

Chapter 8.2: Trigonometric Integrals

Outline For Today

• Integration of sinm(x)cosn(x)

• Integration of secm(x)tann(x)

• Integration of sin(mx)sin(nx)

• Integration of sin(mx)cos(nx)

• Integration of cos(mx)cos(nx)

Integrate sin2k+1(x)cosm(x)

Use sin2(x) = 1-cos2(x) to get

dxxxxdxxx kmmk )sin())(cos1)((cos)(cos)(sin 212

Integrating sin2k+1(x)cosm(x) Continued

Then use u = cos(x)

du = -sin(x) dx

To get

duuudxxxx kmkm )1()sin())(cos1)((cos 22

Lets Do An Example

Lets find

dxxxxdxxx )sin())(cos1)((cos)(sin)(cos 2232

so if we let u = cos(x)

du=-sin(x)dx

dxxx )(cos)(sin 23

Cxx

Cuu

duuudxxxx

5/)cos(3/)cos(

5/3/

)1()sin())(cos1)((cos

53

53

2222

Integrate sinm(x)cos2k+1(x)

Use cos2(x) = 1-sin2(x) to get

dxxxxdxxx kmkm )cos())(sin1)((sin)(cos)(sin 212

Integrating sinm(x)cos2k+1(x) Continued

Then use u = sin(x)

du = cos(x) dx

To get

duuudxxxx kmkm )1()cos())(sin1)((sin 22

Now You Try One

What is ?

A)

dxx)(sin5

Cxxx )(cos)(cos3

2)cos( 53 D) Cxxx )(sin)(sin

3

1)sin( 53

B) Cxxx )(sin)(sin3

2)sin( 53 E) Cxxx )(sin

5

1)(sin

3

1)sin( 53

C) Cxxx )(cos)(cos)cos( 53 F) Cxxx )(cos5

4)(cos

3

2)cos( 53

Now You Try One

What is ?

A)

dxx)(sin5

Cxxx )(cos)(cos3

2)cos( 53 D) Cxxx )(sin)(sin

3

1)sin( 53

B) Cxxx )(sin)(sin3

2)sin( 53 E) Cxxx )(sin

5

1)(sin

3

1)sin( 53

C) Cxxx )(cos)(cos)cos( 53 F) Cxxx )(cos5

4)(cos

3

2)cos( 53

Integrate sin2m(x)cos2n(x)

Use the identities

cos2(x) = 1/2(1+cos(2x))sin2(x) = 1/2(1-cos(2x))

sin(x)cos(x) = 1/2(sin(2x))

Another Example

What is ?

A)

dxxx )(cos)(sin 22

Cxx )]2sin(2[8

1D) Cxx )]2cos(2[

8

1

B) Cxx )]4sin(4[32

1E) Cxx )]2sin([

4

3

C) Cxx )]4cos(4[32

1F) Cxx )]2cos(2[

8

1

Another Example

What is ?

A)

dxxx )(cos)(sin 22

Cxx )]2sin(2[8

1D) Cxx )]2cos(2[

8

1

B) Cxx )]4sin(4[32

1E) Cxx )]2sin([

4

3

C) Cxx )]4cos(4[32

1F) Cxx )]2cos(2[

8

1

Integrate sec2k+2(x)tanm(x)

Use sec2(x) = 1+tan2(x) to get

dxxxxdxxx kmmk )(sec))(tan1)((tan)(tan)(sec 2222

Integrating sec2k+2(x)tanm(x) Continued

Then use u = tan(x)

du = sec2 (x) dx

To get

duuudxxxx kmkm )1()(sec))(tan1)((tan 222

Integrate secm+1(x)tan2k+1(x)

Use tan2(x) = sec2(x)-1 to get

dxxxxxdxxx kmkm )tan()sec()1)()(sec(sec)(tan)(sec 2121

Integrating secm+1(x)tan2k+1(x) Continued

Then use u = sec(x)

du = sec(x)tan(x) dx

To get

duuudxxxxx kmkm )1()tan()sec()1)()(sec(sec 22

Another Example

What is ?

A)

dxxx )(sec)(tan 42

Cxx )(sec5

1)(sec

3

1 53 D)

B) E)

C) F)

Cxx )(sec4

1)(sec

2

1 22

Cxx )(tan5

1)(tan

3

1 53

Cxx )(sec)(tan 53

Cxx )(tan5

1)(sec

3

1 53

Cxx )(tan5

1)(tan

3

1 53

Another Example

What is ?

A)

dxxx )(sec)(tan 42

Cxx )(sec5

1)(sec

3

1 53 D)

B) E)

C) F)

Cxx )(sec4

1)(sec

2

1 22

Cxx )(tan5

1)(tan

3

1 53

Cxx )(sec)(tan 53

Cxx )(tan5

1)(sec

3

1 53

Cxx )(tan5

1)(tan

3

1 53

The Other Cases Are Less Clear

dxx)tan(To find Use u = cos(x)

du = -sin(x)dx

Cx

Cu

Cu

duu

dxx

xdxx

|)sec(|ln

|1

|ln

||ln

1

)cos(

)sin()tan(To get

Integral of sec(x)

This requires a trick:

dxxx

xxx

dxxx

xxxdxx

)tan()sec(

)tan()sec()(sec

)tan()sec(

)tan()sec()sec()sec(

2

Integral of sec(x) Continued

Cxx

Cu

duu

dxxx

xxx

|)tan()sec(|ln

||ln

1

)tan()sec(

)tan()sec()(sec2

We then use u = sec(x) + tan(x)

du = sec2(x)+tan(x)dx

to get :

Integral of sec3(x)

First we integrate by parts with:u = sec(x) dv=sec2(x)dxdu=sec(x)tan(x) v=tan(x)

dxxdxxxx

dxxxxx

dxxxxxdxx

)sec()(sec)tan()sec(

)sec()1)((sec)tan()sec(

)sec()(tan)tan()sec()(sec

3

2

23

Integral of sec3(x) Continued

Cxxxx

dxxxxdxx

|)tan()sec(|ln)tan()sec(

)sec()tan()sec()(sec2 3

So

and we therefore get:

Cxxxxdxx |))tan()sec(|ln)tan()(sec(2

1)(sec3

Other Trig Identities

)]sin()[sin(2

1)cos()sin( BABABA

)]cos()[cos(2

1)sin()sin( BABABA

)]cos()[cos(2

1)cos()cos( BABABA

Other Trig Identities Continued

dxnxmxnxmxdxnxmx )]sin()[sin(2

1)cos()sin(

dxnxmxnxmxdxnxmx )]cos()[cos(2

1)sin()sin(

dxnxmxnxmxdxnxmx )]cos()[cos(2

1)cos()cos(

We can use these identities to get:

Yet Another Example

What is ?

A)

dxxx )cos()2sin(

Cxx )cos()3cos(3

1D)

B) E)

C) F)

Cxx )sin(2

1)3sin(

6

1

Cxx )cos(2

1)3cos(

6

1

Cxx )cos(2

1)3cos(

6

1

Cxx )cos(2

1)3cos(

6

1

Cxx )sin()3cos(

Yet Another Example

What is ?

A)

dxxx )cos()2sin(

Cxx )cos()3cos(3

1D)

B) E)

C) F)

Cxx )sin(2

1)3sin(

6

1

Cxx )cos(2

1)3cos(

6

1

Cxx )cos(2

1)3cos(

6

1

Cxx )cos(2

1)3cos(

6

1

Cxx )sin()3cos(