Math  104  -­‐  Yu  

Math  104  –  Calculus  8.3  Trigonometric  Integrals  

Math  104  -­‐  Yu  

Three  types  of  integrals          

Nicolas Fraiman Math 104

Three types of integrals1.

!

2.

!

3.

Zsin

mx · cosn x dx

Ztanm x · secn x dx

Zsin(mx) sin(nx) dx

Zcos(mx) cos(nx) dx

Zsin(mx) cos(nx) dx

Math  104  -­‐  Yu  

Products  of  Sine  and  Cosine                                                                            (m,n  are  posiEve  integers)    a)  m,  n:  one  or  both  =  1  Use  u-­‐subsEtuEon.  Let  u  =  the  trig  funcEon  with  power  ≠  1.  b) m,  n:  one  (or  both)  odd  (both  greater  than  1)  

1.  Separate  one  factor  from  the  odd  exponent.  2. Use                                                                              to  transform  the  remaining  even  power  into  

the  other  trig  funcEon.  3. Use  u-­‐subsEtuEon  to  finish  the  problem  (let  u  =  the  “other”  trig  

funcEon)  c)  m,  n:  both  even            Replace  all  even  powers  using  the  double-­‐angle  formula:       Nicolas Fraiman

Math 104

Products of sine and cosineIntegrals of the form:

A) m or n odd 1. Separate one factor from the odd exponent. 2. Use to transform the remaining even power into the other function. 3. Use substitution to finish the problem.

B) m and n even Replace all powers using the double-angle formulas

Zsin

mx · cosn x dx

sin

2x+ cos

2x = 1

sin

2x =

12 (1� cos 2x) and cos

2x =

12 (1 + cos 2x)

Nicolas Fraiman Math 104

Products of sine and cosineIntegrals of the form:

A) m or n odd 1. Separate one factor from the odd exponent. 2. Use to transform the remaining even power into the other function. 3. Use substitution to finish the problem.

B) m and n even Replace all powers using the double-angle formulas

Zsin

mx · cosn x dx

sin

2x+ cos

2x = 1

sin

2x =

12 (1� cos 2x) and cos

2x =

12 (1 + cos 2x)

Nicolas Fraiman Math 104

Products of sine and cosineIntegrals of the form:

A) m or n odd 1. Separate one factor from the odd exponent. 2. Use to transform the remaining even power into the other function. 3. Use substitution to finish the problem.

B) m and n even Replace all powers using the double-angle formulas

Zsin

mx · cosn x dx

sin

2x+ cos

2x = 1

sin

2x =

12 (1� cos 2x) and cos

2x =

12 (1 + cos 2x)

Math  104  -­‐  Yu  

Example  1.    

Nicolas Fraiman Math 104

Examples1. Evaluate 2. Evaluate 3. Evaluate

!

4. Evaluate

Zcos

5x sin

2x dx

Z ⇡

0cos

4x sin

2x dx

Z ⇡

0cos

10x sinx dx

Zsin

7x cos

3x dx

Math  104  -­‐  Yu  

Example        

Nicolas Fraiman Math 104

Examples1. Evaluate 2. Evaluate 3. Evaluate

!

4. Evaluate

Zcos

5x sin

2x dx

Z ⇡

0cos

4x sin

2x dx

Z ⇡

0cos

10x sinx dx

Zsin

7x cos

3x dx

Math  104  -­‐  Yu  

Example  3.    

Nicolas Fraiman Math 104

Examples1. Evaluate 2. Evaluate 3. Evaluate

!

4. Evaluate

Zcos

5x sin

2x dx

Z ⇡

0cos

4x sin

2x dx

Z ⇡

0cos

10x sinx dx

Zsin

7x cos

3x dx

Math  104  -­‐  Yu  

Products  of  tangent  and  secant                                                                                    

c) Other cases: there is no set method...

b) n even

1. Factor out (sec

2x) which is the derivative of tanx.

2. Use sec

2x = tan

2x+1 to transform the remaining even power of secx

into tanx

3. Use u-substitution to finish the problem (u = tanx)

2/11/2014

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Math 104 – Rimmer8.2 Integrating Powers of Trig. Functions ( )tan sec , positive integersm nx x dx m n∫

) ( )B the power of sec :n x even 2 4: tan secex x xdx∫21. Factor out sec x

2 4 2 2 2: tan sec t sec sn ca eex x x xx x=

2 22. If 2, use sec 1 tan to transform the remaining even power

of sec to be in terms of tan

n x x

x x

> = +

( )2 2 2 2 2 2: tan sec sec sec1t n tanaex x x x x xx= +

( )3. use substitution to finish the problem let tanu u x! =

( )22 24 2: tan sec tan 1 tan s ecxex x xdx x dxx+=∫ ∫2

tan

sec

u x

du xdx

=

=( )2 21u u du= +∫ ( )2 4u u du= +∫

3 5

3 5u u C= + +

3 51 13 5

tan tanx x C= + +

Math 104 – Rimmer8.2 Integrating Powers of Trig. Functions tan sec m nx x dx∫

)C For all other cases, there is no set method Here are some examples:

: tanex xdx∫sin

cos

xdx

x= ∫

1 du

u

!∫

cos

sin

u x

du xdx

=

= !

ln u C= ! +

ln cos x C= ! +

1ln cos x C

!= +

tan ln secxdx x C= +∫

: secex xdx∫convenient version of 1

sec tansec

sec tan

x xx dx

x x

+= "

+∫2sec sec tan

sec tan

x x xdx

x x

+=

+∫

( )2

sec tan

sec tan sec

u x x

du x x x dx

= +

= +sec ln sec tanxdx x x C= + +∫1

duu∫ ln u C= +

Rtanm x · secn xdx

a) m odd, n positive

1. Factor out (tanx secx), which is the derivative of secx.

2. Use tan

2x = sec

2x � 1 to transform the remaining even power of

tanx into secx.

3. Use u-substitution to finish the problem (u = secx)

Math  104  -­‐  Yu  

Example  4.    

Nicolas Fraiman Math 104

Examples5. Evaluate 6. Evaluate

Ztan3 x sec3 x dx

Ztan2 x sec4 x dx

Math  104  -­‐  Yu  

Example  5.              

Nicolas Fraiman Math 104

Examples5. Evaluate 6. Evaluate

Ztan3 x sec3 x dx

Ztan2 x sec4 x dx

Math  104  -­‐  Yu  

Example  Other  cases:  6.      Evaluate  

7.  Evaluate    

Nicolas Fraiman Math 104

ExamplesFor the remaining cases (m even, n odd) there is no method.

7. Evaluate and

Ztanx dx

Zsecx dx.

Nicolas Fraiman Math 104

ExamplesFor the remaining cases (m even, n odd) there is no method.

7. Evaluate and

Ztanx dx

Zsecx dx.

Math  104  -­‐  Yu  

Linear  factors  of  sine  and  cosine    a)  If  m≠n,  change  the  product  into  a  sum  using  the  following  

idenEEes:    

b)  If  m=n,  the  formulas  above  become  double-­‐angle  formulas  

2/11/2014

8

Math 104 – Rimmer8.2 Integrating Powers of Trig. Functions

( ) ( )3. sin sinmx nx dx∫ ( ) ( )cos cosmx nx dx∫ ( ) ( )sin cosmx nx dx∫, rational with m n m n!

We change the product into a sum using the following identities:

( ) ( ) [ ]( ) [ ]( )1

sin sin cos cos2

mx nx m n x m n x = " + "

( ) ( ) [ ]( ) [ ]( )1

cos cos cos cos2

mx nx m n x m n x = " + +

( ) ( ) [ ]( ) [ ]( )1

sin cos sin sin2

mx nx m n x m n x = " + +

( ) ( )sin 3 cos 5x x dx∫ [ ]( ) [ ]( )1

sin 3 5 sin 3 52

x x dx = " + ∫ +

( ) ( )1

sin 2 sin 82

x x dx= " + ∫ ( ) ( )1

sin 8 sin 22

x x dx= " ∫( ) ( )1 1

8 2

1cos 8 cos 2

2x x C= " + + ( ) ( )1 1

16 4cos 8 cos 2x x C= " + +

2/11/2014

8

Math 104 – Rimmer8.2 Integrating Powers of Trig. Functions

( ) ( )3. sin sinmx nx dx∫ ( ) ( )cos cosmx nx dx∫ ( ) ( )sin cosmx nx dx∫, rational with m n m n!

We change the product into a sum using the following identities:

( ) ( ) [ ]( ) [ ]( )1

sin sin cos cos2

mx nx m n x m n x = " + "

( ) ( ) [ ]( ) [ ]( )1

cos cos cos cos2

mx nx m n x m n x = " + +

( ) ( ) [ ]( ) [ ]( )1

sin cos sin sin2

mx nx m n x m n x = " + +

( ) ( )sin 3 cos 5x x dx∫ [ ]( ) [ ]( )1

sin 3 5 sin 3 52

x x dx = " + ∫ +

( ) ( )1

sin 2 sin 82

x x dx= " + ∫ ( ) ( )1

sin 8 sin 22

x x dx= " ∫( ) ( )1 1

8 2

1cos 8 cos 2

2x x C= " + + ( ) ( )1 1

16 4cos 8 cos 2x x C= " + +

Math  104  -­‐  Yu  

Example  8.    

Nicolas Fraiman Math 104

Examples!

8. Evaluate Z

sin(3x) cos(5x) dx