Section 8.3 – Trigonometric
Integrals
Rewriting tools for Integrating:
Trigonometric Identities
Use these identities to help evaluate integrals with trigonometric functions.
sin x 
1
csc x
csc x 
1
sin x
tan x 
sec x 
1
cos x
tan x 
1
cot x
cos x 
1
sec x
cot x 
1
tan x
sin x
cos x
cot x 
cos x
sin x
Rewriting tools for Integrating: Pythagorean
Identities
Use these identities to help evaluate integrals with trigonometric functions.
sin x  cos x  1
2
2
1  tan x  sec x
2
2
1  cot x  csc x
2
2
Rewriting tools for Integrating:
Double Angle Formulas
Use these identities to help evaluate integrals with trigonometric functions.
sin 2 x  2 cos x sin x
cos 2 x  cos x  sin x  2 cos x  1  1  2sin x
2
2
2
tan x
tan 2 x  12tan
2
x
2
Rewriting tools for Integrating:
Power-Reducing Formulas
Use these identities to help evaluate integrals with trigonometric functions.
sin u 
1 cos  2 u 
2
cos u 
1 cos  2 u 
2
2
2
1 cos  2 u 
tan u  1 cos 2u 
2
Goal of Trigonometric Integrals
If needed, rewrite
the integral into a
form that can be
integrated.
Example 1
Evaluate:
sin3 𝑥 𝑑𝑥
sin
x
dx

sin
x
sin
x
dx


3
2
  1  cos x  sin x dx
2
   sin x  cos x sin x  dx
2
  sin x dx   cos x sin x dx
2
  cos x  cos x  C
1
3
3
Example 2
Evaluate:
 sin
4
sin4 𝑥 cos 5 𝑥 𝑑𝑥
x cos x dx   sin x  cos x  cos x dx
5
4
2
2
  sin x 1  sin x  cos x dx
4
2
2
  sin 4 x 1  2sin 2 x  sin 4 x  cos x dx
   sin 4 x cos x  2sin 6 x cos  sin 8 x cos x  dx
  sin x cos x dx  2  sin x cos dx   sin x cos x dx
4
6
 15 sin 5 x  72 sin 7 x  19 sin 9 x  C
8
Example 3
Evaluate:
 sin
4
sin4 𝑥 𝑑𝑥
x dx    sin x  dx   
2
2



1 cos 2 x 2
2
1 2cos 2 x  cos 2 2 x
4
dx
 dx
 14   1 dx  2  cos 2 x dx   cos 2 2 x dx 


 14  x  sin 2 x   1 cos2 4 x dx 


 14  x  sin 2 x   12 dx  12  cos 4 x dx 


 14  x  sin 2 x  12 x  81 sin 4 x 
 14  23 x  sin 2 x  81 sin 4 x   C
Example 4
sin2 𝑥 cos 4 𝑥 𝑑𝑥
x dx   
  cos x  dx
Evaluate:
2
4
sin
x
cos

1 cos 2 x
2
2
2
   1cos2 2 x  1 cos2 2 x  dx
2
   1cos2 2 x 

1 2cos 2 x  cos 2 2 x
4
  1 2cos 2 x  cos
2
 dx
2 x  cos 2 x  2cos 2 2 x  cos3 2 x
8
dx
 18  1  cos 2 x  cos 2 2 x  cos 3 2 x  dx
 18  1  cos 2 x  1 cos2 4 x  cos 2 2 x cos 2 x  dx


 18  1  cos 2 x  12  12 cos 4 x  1  sin 2 2 x  cos 2 x dx
 18   12  cos 2 x  12 cos 4 x  cos 2 x  sin 2 2 x cos 2 x  dx
 18   12  12 cos 4 x  sin 2 2 x cos 2 x  dx
 18  12 x  81 sin 4 x  16 sin 3 2 x   C
White Board Challenge
Combine Integration by Parts and
Trigonometric Identities to evaluate: sec 3 𝑥 𝑑𝑥
2
Pick the u and dv.
dv  sec x dx
u  sec x
Find du and v. du  sec x tan x dx
v  tan x
Apply the
3
2
sec
x
dx


sec
x
tan
x dx
sec
x
tan
x
formula. 

 sec
 sec
2  sec
 sec
3
3
x dx  sec x tan x   sec x  sec x  1 dx
2
x dx  sec x tan x   sec x dx   sec x dx
3
3
x dx  sec x tan x  ln sec x  tan x  C
3
x dx  12 sec x tan x  12 ln sec x  tan x  C
White Board Challenge
White Board Challenge
Evaluate: sec 3 𝑥 𝑑𝑥
3
2
 sec x dx   sec x sec x dx
 sec
 sec
 sec
3
3
x dx   sec x 1  tan x  dx
2
x dx   sec x dx   sec x tan x dx
2
x dx  ln sec x  tan x   tan x  sec x tan x dx
Pick the u and dv.
dv  sec x tan x dx
u  tan x
2
Find du and v. du  sec x dx
v  sec x
3
3
sec
x
dx

ln
sec
x

tan
x

sec
x
tan
x

sec
 3
 x dx
2  sec x dx  sec x tan x  ln sec x  tan x  C
3
3
1
1
sec
x
dx

ln
sec
x

tan
x

2
2 sec x tan x  C

Example 5
sec 5 𝑥 𝑑𝑥
u  sec3 x
du  3sec 2 x sec x tan x dx
Evaluate:
Pick the u and dv.
Find du and v.
dv  sec 2 x dx
v  tan x
 sec x dx  sec x tan x   3sec x tan x dx
5
 sec
 sec
5
5
3
3
2
Apply the
formula.
x dx  sec x tan x  3 sec x  sec x  1 dx
3
3
2
x dx  sec3 x tan x  3 sec5 x dx  3 sec 3 x dx
4  sec x dx  sec x tan x  3  12 sec x tan x  12 ln sec x  tan x  C 
5
3
3
3
sec
x
dx

sec
x
tan
x

sec
x
tan
x

8
8 ln sec x  tan x  C

5
1
4
3
Example 6
Evaluate:
tan2 𝑥 sec 3 𝑥 𝑑𝑥
2
3
2
3
tan
x
sec
x
dx

sec
x

1
sec
dx




   sec5 x  sec3 x  dx
  sec5 x dx   sec3 x dx
  14 sec3 x tan x  83 sec x tan x  83 ln sec x  tan x    12 sec x tan x  12 ln sec x  tan x   C
 14 sec3 x tan x  18 sec x tan x  18 ln sec x  tan x  C
Example 7
Evaluate:

1 tan 2 x
sec2 x
1−tan2 𝑥
𝑑𝑥
2
sec 𝑥
dx   sec12 x 1  tan 2 x  dx


sin x
  cos 2 x 1  cos
dx
2
x
2
   cos 2 x  sin 2 x  dx
  cos 2 x dx   sin 2 x dx
  1 cos2 2 x dx   1cos2 2 x dx
  12 dx  12  cos 2 x dx 

1
2
dx  12  cos 2 x dx
  12 dx  12  cos 2 x dx   12 dx  12  cos 2 x dx
  cos  2x  dx
 12 sin  2x   C

Example 7
Example 7
Evaluate:

1 tan 2 x
sec2 x
1−tan2 𝑥
𝑑𝑥
2
sec 𝑥
dx   sec12 x 1  tan 2 x  dx


sin x
  cos 2 x 1  cos
dx
2
x
2
   cos 2 x  sin 2 x  dx
  cos  2x  dx
 12 sin  2x   C
Example 8
𝑑𝑥
cos 𝑥−1
Evaluate:

dx
cos x 1
cos x 1
  cosdxx 1  cos
x 1
cos x 1
  cos
dx
2
x 1
  cossinx2x1 dx
 

cos x
sin 2 x

 sin12 x dx
2
x
1
    cos


csc
x  dx
sin x sin x


2
cot
x

csc
x
dx

csc

 x dx
    csc x  cot x   C
 csc x  cot x  C

Example 9
Evaluate:
Substitution:
cos2 sin 𝑥 cos 𝑥 𝑑𝑥
u  sin x du  cos x dx
2
2
cos
sin
x
cos
x
dx

cos
  
 u du

1 cos 2 u 
2
du
  12 du  12  cos 2u du
 12 u  14 cos 2u  C
 12 sin x  14 sin  2sin x   C
Try to still follow the rules even if m and n are negative AND count
zero as even.
General Guidelines for Trig Integrals
For sin𝑛 𝑥 cos 𝑚 𝑥 𝑑𝑥 we have the following:
1. n odd.
Strip one sine out and convert the rest of the integral to
only cosines using sin2 𝑥 = 1 − cos 2 𝑥. Then use the
substitution with 𝑢 = cos 𝑥.
2. m odd.
Strip one cosine out and convert the rest of the integral to
only sines using cos 2 𝑥 = 1 − sin2 𝑥. Then use the
substitution with 𝑢 = sin 𝑥.
3. n and m both odd.
Use either 1. or 2.
4. n and m both even.
Use double angle, half angle, and/or power reducing
formulas to reduce the integral into a form that can be
integrated.
Try to still follow the rules even if m and n are negative AND count
zero as even.
General Guidelines for Trig Integrals
For
1.
2.
3.
4.
tan𝑛 𝑥 sec 𝑚 𝑥 𝑑𝑥 we have the following:
n odd.
Strip one tangent and one secant out and convert the
rest of the integral to only secants using tan2 𝑥 = sec 2 𝑥 − 1.
Then use the substitution with 𝑢 = sec 𝑥.
m even.
Strip two secants out and convert the rest of the integral to
only tangents using sec 2 𝑥 = 1 + tan2 𝑥. Then use the
substitution with 𝑢 = tan 𝑥.
n odd and m even.
Use either 1. or 2.
n even and m odd.
Each integral will be dealt with differently. Try Trigonometric
Identities, Substitution, or Integration by Parts.