MATH 155/GRACEY
WORKSHEET/8.3
NAME _________________________
TRIGONOMETRIC INTEGRALS
We will study techniques for evaluating integral of the form
m
n
sin
x
cos
xdx

and
m
n
sec
x
tan
xdx

GUIDELINES FOR EVALUATING INTEGRALS INVOLVING SINE AND
COSINE
1. If the power of the sine is odd and positive, save one sine factor and
convert the remaining factors to cosines. Then expand and integrate.
convert to cosines
using pythagorean
identity
 sin
odd

2 k 1
x cos n xdx  
save for du






k
2
n
1
 sin x  cos x sin xdx
k
  1  cos x  cos n x sin xdx
2
2. If the power of the cosine is odd and positive, save one cosine
factor and convert the remaining factors to sines. Then expand and
integrate.
convert to sines
using pythagorean
identity
m
sin
 x cos
odd

2 k 1
save for du


 


k
xdx   sin m x  cos 2 x  cos1 xdx
k
  sin m x 1  sin 2 x  cos xdx
3. If the powers of both the sine and cosine are even and nonnegative,
make repeated use of the identities
sin 2 x 
1  cos 2 x
2
and
cos 2 x 
1  cos 2 x
2
To convert the integrand to odd powers of the cosine. Then proceed
as in guideline 2.
GUIDELINES FOR EVALUATING INTEGRALS INVOLVING SECANT
AND TANGENT
1. If the power of the secant is even and positive, save one secantsquared factor and convert the remaining factors to tangents. Then
expand and integrate.
convert to tangents
using pythagorean
identity
even

2k
sec

save for du


 

k

1
x tan n xdx    sec 2 x  tan n x sec 2 xdx
  1  tan x 
2
k 1
tan n x sec 2 xdx
2. If the power of the tangent is odd and positive, save one secanttangent factor and convert the remaining factors to secants. Then
expand and integrate.
convert to secants
using pythagorean
identity
m
sec
 x tan
odd

2 k 1
save for du
 


k
xdx   secm 1 x  tan 2 x  sec1 x tan xdx
k
  sec x  sec x  1 sec x tan xdx
m
2
3. If there are no secant factors and the power of the tangent is even
and positive, convert a tangent-squared factor to a secant-squared
factor, then expand and repeat, if necessary.
convert to secants
using pythagorean
identity



n
n2
2
tan
xdx

tan
x
tan
x  dx



  tan n  2 x  sec 2 x  1 dx
4. If the integral is of the form
 sec
m
xdx , where m is odd and
positive, use integration by parts!
5. If none of these techniques work, try converting to sines and cosines.
INTEGRALS INVOLVING SINE-COSINE PRODUCTS WITH
DIFFERENT ANGLES
1
cos  m  n  x   cos  m  n  x 
2
1
sin  mx  cos  nx   sin  m  n  x   sin  m  n  x 
2
1
cos  mx  cos  nx   cos  m  n  x   cos  m  n  x 
2
1
cos  mx  sin  nx   sin  m  n  x   sin  m  n  x 
2
sin  mx  sin  nx  






1. Integrate.
a.
7
cos
 xdx


b.
4
sin
 3xdx
c.
5
sec
 xdx
d.
5
tan
 2 x sec 2 xdx
e.
4
2
tan
x
sec
xdx

f.
g.
 sin  3x  cos  2 x  dx
6 x
tan
  2  dx