Transcript Trigonometric Integrals

SEC 8.2: TRIGONOMETRIC INTEGRALS
Example
Find  cos x dx
3
3
2
cos
x
dx

cos
x cos xdx


  (1  sin 2 x) cos xdx
Example
Find
5
2
sin
x
cos
x dx

5
2
4
2
sin
x
cos
x
dx

sin
x
cos
x sin x dx


  (1  cos 2 x) 2 cos 2 x sin xdx
TRIGONOMETRIC INTEGRALS
 sin
m
n
x cos x dx
sin  odd
m is odd
cos  odd
n is odd
save one sin
1 save one cos with dx
1
2 use cos x  1- sin x
2 use sin x  1- cos x
2
2
to express the remaining
factors in terms of sin
3
2
cos
x
dx

cos
x cos xdx


2
2
to express the remaining
factors in terms of cos
5
2
sin
x
cos
x dx

  sin 4 x cos 2 x sin x dx
TRIGONOMETRIC INTEGRALS
odd
cos
even
even
odd
odd
odd
even
even
sin
sin  even
cos  even
4
sin
 dx

1
4
2


1

cos
2
x
dx

1
use half angle
sin 2 x  12 (1- cos 2 x)
cos 2 x  12 (1  cos 2 x)
2
sometimes helpful to use
sin x cos x  12 sin 2 x
TRIGONOMETRIC INTEGRALS
Eliminating Square Roots
we use the identity
cos 2 x  12 (1  cos 2 x)
to eliminate a square root.
Example
Find


4
0
1  cos 4 x dx
TRIGONOMETRIC INTEGRALS
We can use a similar strategy to
evaluate integrals of the form
 tan
m
n
x sec x dx
Example
Find
 tan
6
4
x sec x dx
u  tan x  du  sec 2 xdx
sec 2 x  1  tan 2 x
Example
Find
5
4
tan
x
sec
x dx

u  sec x  du  sec x tan xdx
tan 2 x  1  sec 2 x
TRIGONOMETRIC INTEGRALS
 tan
m
n
x sec x dx
tan  odd
m is odd
sec  even
n is even
1 save one sec 2
2 use sec x  1  tan x
2
to express the remaining
factors in terms of tan
2
1
save one sec x tan x
2
use tan 2 x  sec 2 x  1
to express the remaining
factors in terms of sec
EXAM-2
Term-082
TRIGONOMETRIC INTEGRALS
tan
even
tan
sec
odd
even
even
odd
odd
odd
even
even
sec
odd
the guidelines are not as clear-cut. We may need to use identities,
integration by parts, and occasionally a little ingenuity.
TRIGONOMETRIC INTEGRALS
tan
even
sec
odd
Example
Find
3
sec
 xdx
the guidelines are not as clear-cut. We may
need to use identities, integration by parts, and
occasionally a little ingenuity.
If an even power of tangent appears with an
odd power of secant, it is helpful to express
the integrand completely in terms of sec x
Powers of sec x may require integration by
parts, as shown in the following example.
TRIGONOMETRIC INTEGRALS
Example
Find
3
sec
xdx

TRIGONOMETRIC INTEGRALS
REMARK
Integrals of the form
m
n
cot
x
csc
x dx

can be found by similar methods because
of the identity
1  cot 2 x  csc 2 x
cot
x
csc
x
dx

m
n
cot  odd
m is odd
csc  even
n is even
1 save one sec 2
2
csc x  1  cot x
2
2
to express the remaining
factors in terms of cot
1
save one csc x cot x
2
use cot 2 x  csc 2 x  1
to express the remaining
factors in terms of csc
EXAM-2
Term-122
Product of Sines and Cosines
 cos mx cos nx dx
 sin mx sin nx dx
 sin mx cos nx
dx
EXAM-2 Term-122
TRIGONOMETRIC INTEGRALS
Powers of Sines
and Cosines
Products of
Sines and
Cosines
TRIGONOMETRIC
INTEGRALS
Powers of tan x
and sec x
Eliminating
Square
Roots
EXAM-2
Term-092
EXAM-2
Term-092
TRIGONOMETRIC INTEGRALS
function of tan and sec

function of Sines and Cosines
xdx


 f (cos x)sin
2
f (tan x)sec

xdx


du
du
x tan
xdx

 f (sec x)sec
xdx


 f (sin x)cos
du
du
TRIGONOMETRIC
INTEGRALS
function of cot and csc

2
f (cot x)csc

xdx


du
x cot
xdx

 f (csc x)csc
du
EXAM-2
Term-092
EXAM-2
Term-092
EXAM-2
Term-092
EXAM-2
Term-092