Mat h 12 06 Ca lc ul us – Sec . 7. 2: T ri gono met ric Inte gral s
I.
Eval uati ng
! sin
m
x cosn x dx
A. Strategy for Evaluating
! sin
m
n
x cos x dx
1. If the power of cosine is odd
(n = 2k + 1) , save one cosine factor and use
cos2 x = 1! sin2 x to express the remaining factors in terms of sine:
! sin
m
x cos2k +1 x dx = ! sin m x cos2k x cos x dx
( )
x (1 " sin x )
k
= ! sinm x cos2 x cos x dx
= ! sinm
2
k
cos x dx
Then substitute u= sinx.
2. If the power of sine is odd
(m = 2k + 1) , save one
sine factor and use
sin2 x = 1 ! cos2 x to express the remaining factors in terms of cosine:
! sin
2k +1
x cosn x dx = ! sin 2k x cos n x sin x dx
( )
= ! (1 " cos x ) cos
k
= ! sin 2 x cosn x sin x dx
2
k
n
x sin x dx
Then substitute u= cosx.
3. If the powers of both sine and cosine are odd, either 1 or 2 can be used.
(Easier problem if you factor out the term with the smaller exponent.)
4. If the powers of both sine and cosine are even, use the half-angle
2
identities sin
x = 12 (1! cos2x )
cos2 x =
5. It is sometimes helpful to use the identity
B. Examples -- Evaluate the following
1.
" sin ! cos ! d!
4
1
2
(1 + cos2x )
sin x cos x = 12 sin2x
2.
3.
" sin
!
3
! d!
sin 2 x cos3 x
x
dx
4. Find the area bounded by the curve of
4
y = sin (3x ) and the x-axis from x = 0 to
x = !3 .
II.
Eval uati ng
! tan
m
x sec n x dx
A. Strategy for Evaluating
! tan
m
x sec n x dx
1. If the power of secant is even
2
(n = 2k ) , save a factor of sec x and use
2
2
sec x = 1 + tan x to express the remaining factors in terms of tangent:
" tan
m
x sec 2k x dx = " tan m x sec 2k !2 x sec2 x dx
( ) sec x dx
x (1 + tan x ) sec x dx
= " tan m x sec 2 x
k !1
= " tan m
2
Then substitute u= tanx.
2
k !1
2
2. If the power of tangent is odd
(m = 2k + 1) , save a factor of secxtanx and use
tan2 x = sec 2 x ! 1 to express the remaining factors in terms of secant:
" tan
2k +1
x sec n x dx = " tan2k x sec n!1 x sec x tan x dx
( )
= " (sec x ! 1) sec
k
= " tan2 x sec n!1 x sec x tan x dx
2
Then substitute u= secx.
B. Examples -- Evaluate the following
1.
! tan
6
"
2.
y dy
6
# tan !
0
sec3 ! d!
k
n!1
x sec x tan x dx
3. Find the volume of the solid generated by revolving the region bounded by the curve
of
III.
( )
( )
y = sec2 x 2 tan 2 x 2 , the x-axis and the line x =
Eval uati ng
!
2
about the y-axis.
! tan x dx , ! cot x dx , ! sec x dx , ! csc x dx
A. Strategy for Evaluating
1. When evaluating
! tan x dx , ! cot x dx , ! sec x dx , ! csc x dx
! tan x dx
or
! cot x dx
rewrite the integrand in terms of sine and
cosine and then use u-substitution.
2. When evaluating
sec x + tan x
! sec x dx multiply the integrand by the form of one, sec x + tan x
,
and then use u-substitution.
3. When evaluating
csc x + cot x
! csc x dx multiply the integrand by the form of one, csc x + cot x
and then use u-substitution.
,
B. Examples
"
1.
#
"
2.
2
cot ! d!
4
! sec x dx
C. Integral Formulas for tan(x), cot(x), sec(x), csc(x)
1.
2.
3.
4.
! tan u du =
! cot u du =
! secu du =
! csc u du =
- ln cosu + C = ln sec u + C
ln sinu + C = - ln cscu + C
ln secu + tan u + C
- ln cscu + cot u + C