Download 6.2 (part 1)

6.2 Trigonometric Integrals and Substitutions (part 1)
Trigonometric integrals like this:
 sin
m
x cosn xdx
If m is an odd positive integer, let u =
If n is an odd positive integer, let u =
If both m and n are even nonnegative integers, then use the identities:
cos2 x =
An identity that is often used in each of these cases is:
EXAMPLES
cos 3 x
dx
3
sin x
1.

2.
 sin
2
x dx
sin2 x =
sin2x + cos2x = 1
 /2
3.

4.
 sin
0
cos2 3x dx
2
x cos2 x dx
Trigonometric integrals like this:
 tan
m
xsecn xdx
If m is an odd positive integer, then normally let u =
If n is an even positive integer, then normally let u =
If m is an even positive integer and n is an odd positive integer, then a formula
and/or integration by parts should be used.
In any case, the identity 1 + tan2x = sec2x is frequently used.
EXAMPLES
1.
 tan
2.
 tan xsec
3
x sec6 x dx
6
x dx
Integrals of the type
 cot
m
x cscn xdx are handled in the same manner as
The important identity here is 1 + cot2x = csc2x
 tan
m
xsecn xdx .