Download Math 1432 cos sin tan sec cot csc cos cos cos sin sin sin cos cos cos

Math 1432
Bubble in PS ID and Popper Number very carefully. If you make a bubbling
mistake, your scantron can’t be saved in the system. In that case, you will not
get credit for the popper even if you turned it in.
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Be considerate of others in class. Respect your friends and do not distract
anyone during the lecture
Section 8.2 Powers and Products of Trigonometric Functions
Recall the following identities:
cos 2  x   sin 2  x   1
1  tan 2  x   sec 2  x 
1  cot 2  x   csc 2  x 
cos 2  x  
1  cos  2x 
2
sin 2  x  
1  cos  2x 
2
sin  2x   2 sin x cos x
cos  2x   cos 2 x  sin 2 x
1
In this section, we will study techniques for evaluating integrals of the form
 sin
m
x cos x dx

n
and
sec m x tan n dx
where either m or n is a positive integer.
To find antiderivatives for these forms, try to break them into combinations of
trigonometric integrals to which you can apply a formula:

 u n1
C

u n du   n  1
ln u  C

if n  1
if n   1
.
Integrals Involving Powers of Sine and Cosine
1.
If m or n odd:
2
a. m odd: rewrite
identity
sin m x as sin m-1 xsin x (m-1 is even so can use
sin 2 x  1  cos 2 x )
cos m-1 x cos x (n-1 is even so can use
cos 2 x  1  sin 2 x )
m
b. n odd: rewrite cos x as
identity
Example:  sin 3 xdx
Example:
 sin
3
x cos 2 xdx
3
Example:
 cos
5
xdx
Example:  sin 4 x cos 5 xdx
4
2. If m and n even use these identities:
sin 2 x 
1  cos 2x
2
and /or
cos 2 x 
1  cos 2x
2
.
Example:  cos 2 xdx
5
Integrals involving Secants and Tangents
tan 2 x  1  sec 2 x
For
m
n
tan
xsec
xdx
ò
a. n even: rewrite
can use identity
tan m xsecn x as tan m xsecn-2 xsec2 x (then you
sec 2 x  tan 2 x  1 )
tanm-1 xsecn-1 x ×sec x tan x (m-1 is even so
2
2
can use identity tan x = sec x -1)
2
2
m
m even and n odd: rewrite tan x using tan x = sec x -1
b. m odd: rewrite as
c.
Example:
 tan
3
xdx
6
Example:  sec 4 xdx
Example:  sec 4 x tan 2 xdx
7
Example:
sec x
 tan 2 x dx
POPPER#
8
Q#
ò
cos x sin 3 x dx
a.
1
cos2 x + C
2
b.
c.
1 4
sin x + C
4
d. none of these
1
cos 4 x + C
4
9