8/12/2019 MATH 37 UNIT 2.2
1/32
2.2
POWERS OFTRIGONOMETRIC
FUNCTIONS
8/12/2019 MATH 37 UNIT 2.2
2/32
Solvable integral forms . . .
Cxcosxdxsin
Cxsindxcos Cxseclnxdxtan
Cxcsclnxdxcot Cxtanxseclnxdxsec
Cxcotxcsclnxdxcsc
8/12/2019 MATH 37 UNIT 2.2
3/32
Solvable integral forms . . .
Cxtanxdxsec 2
Cxcotxdxcsc
2
Cxsecxdxtanxsec Cxcscxdxcotxcsc
8/12/2019 MATH 37 UNIT 2.2
4/32
Restr ict ions
xdxsinm
xdxcos
n
xdxcosxsin nm
xdxtan
m
xdxcot
n
Form 1.
Form 2.
8/12/2019 MATH 37 UNIT 2.2
5/32
Restr ict ions
Form 3.
Form 4.
xdxsec
m
xdxcsc
m
xdxsectan nm
xdxcscxcot nm
8/12/2019 MATH 37 UNIT 2.2
6/32
Form 1
xdxsinm
xdxcos
n
xdxcosxsin nm
Case 1. m or n is odd .
1. Separate one facto r o f the
odd-powered funct ion.
2. Express the rest in terms o f
the other funct ion using
122 xcosxsin
3. Proceed w ith subst i tut ion .
8/12/2019 MATH 37 UNIT 2.2
7/32
Form 1
xdxsinm
xdxcos
n
xdxcosxsin nm
Case 2. m and n are even.
Use2
212 xcosxsin
2
212 xcosxcos
8/12/2019 MATH 37 UNIT 2.2
8/32
Example.
Evaluate .
xdxsin3
Solut ion:
xdxsin
3
dxxsinxsin
2
dxxsinxcos21
Let .xcosu dxxsindu
dxxsindu
8/12/2019 MATH 37 UNIT 2.2
9/32
Solut ion (cont inued)
xdxsin3
duu21
Cuu
3
3
Cxcosxcos 3
31
8/12/2019 MATH 37 UNIT 2.2
10/32
Example.
Evaluate . xdxcosxsin 32
Answer : Cxsinxsin
53
53
8/12/2019 MATH 37 UNIT 2.2
11/32
Example.
Evaluate .
xdxcosxsin 22
Solut ion:
xdxcosxsin 22
dxxcosxcos 2121
4
1
dxxcos 22141
8/12/2019 MATH 37 UNIT 2.2
12/32
Solut ion (cont inued)
dxxcos 2
2
xdxcosxsin 22
dxxcos
221
4
1
dx
xcos
2
41
dxxcosdx 421
2
1
x2
1Cxsin 4
8
1
8/12/2019 MATH 37 UNIT 2.2
13/32
Solut ion (cont inued)
xdxcosxsin 22
dxxcos 221
4
1
dxxcosdx 22
4
1
4
1
x4
1Cxsinx
4
8
1
2
1
4
1
Cxsinx 432
1
8
1
8/12/2019 MATH 37 UNIT 2.2
14/32
Form 2. xdxtanm xdxcotn
1. Separate or .xtan2
xcot2
2. Express i t in terms o f o rus ing
xsecxcsc
122 xsecxtan
122 xcscxcot
3. Proceed w ith subst i tut ion .
8/12/2019 MATH 37 UNIT 2.2
15/32
Example.
Evaluate .
xdxtan3
Solut ion:
xdxtan3
dxxtanxtan
2
dxxsecxtan 12
dxxtanxsecxtan 2
8/12/2019 MATH 37 UNIT 2.2
16/32
8/12/2019 MATH 37 UNIT 2.2
17/32
Solut ion (cont inued)
Cxseclnxtan
2
2
dxxtanxsecxtan 2
xdxtan3
8/12/2019 MATH 37 UNIT 2.2
18/32
Example.
Evaluate . xdxcot4
Answer : Cxxcotxcot
3
3
8/12/2019 MATH 37 UNIT 2.2
19/32
Form 3.
1. Separate or .xsec2
xcsc2
2. Express the rest in terms o f
or us ingxtan xcot
xtanxsec 22
1xcotxcsc
221
3. Proceed w ith subst i tut ion .
xdxsecm xdxcscm
Case 1. m is even .
8/12/2019 MATH 37 UNIT 2.2
20/32
Use In teg rat ion by Parts w ith
xdxsecdv 2
xdxcscdv
2
Case 2. m is odd .
Form 3. xdxsecm xdxcscm
8/12/2019 MATH 37 UNIT 2.2
21/32
Example.
Evaluate .
xdxcsc4
Solut ion:
dxxcscxcsc
22
dxxcscxcot 221
xdxcsc4
Let .xcotu dxxcscdu 2
8/12/2019 MATH 37 UNIT 2.2
22/32
Solut ion (cont inued)
xdxcsc4
dxxcscxcot 221
duu21
Cu
u
3
3
xcot Cxcot
3
3
8/12/2019 MATH 37 UNIT 2.2
23/32
Example.
Evaluate .
xdxsec3
Solut ion:
xdxsec
3
u dv dxxsec xsec2
du xdxtanxsec v xtan
By IBP!
xdxsec3 dxxsecxtanxtanxsec 2
8/12/2019 MATH 37 UNIT 2.2
24/32
Solut ion (cont inued)
xdxsec
3
dxxsecxtanxtanxsec
2
dxxsecxsecxtanxsec 12
dxxsecdxxsecxtanxsec 3
Hence,
xdxsec32 dxxsecxtanxsec
8/12/2019 MATH 37 UNIT 2.2
25/32
Answer :
xdxsec3
Cxtanxseclnxtanxsec
2
8/12/2019 MATH 37 UNIT 2.2
26/32
Form 4
Case 1. n is even . xdxsectan nm
xdxcscxcot nm
1. Separate or .xsec2
xcsc2
2. Express the rest in terms o f
or us ingxtan x cot
xtanxsec 221
xcotxcsc 221
3. Proceed w ith subst i tut ion .
8/12/2019 MATH 37 UNIT 2.2
27/32
Form 4
Case 2. m is odd . xdxsectan nm
xdxcscxcot nm
2. Express the res t in terms ofor .xsec x csc
3. Proceed w ith subst i tut ion .
1. Separate one facto r o f
or and one facto r
of or .
xtan
xsec
xcot
xcsc
8/12/2019 MATH 37 UNIT 2.2
28/32
Example.
Evaluate .
Solut ion:
xdxsecxtan 42
xdxsecxtan 42
dxxsecxsecxtan 222
dxxsecxtanxtan 222 1
8/12/2019 MATH 37 UNIT 2.2
29/32
Solut ion (cont inued)
Let .xtanu xdxsecdu 2
duuu 22 1 xdxsecxtan 42
duuu 42
C
uu
53
53
Cxtanxtan
53
53
8/12/2019 MATH 37 UNIT 2.2
30/32
Example.
Evaluate .
Answer :
xdxcscxcot 33
Cxcscxcsc
53
53
8/12/2019 MATH 37 UNIT 2.2
31/32
In general . . .
When odd-powered , separate one
factor.
When even-powered , separate twofactors.
What you separate is a derivat ive
(o r di f feren t ial) o f some funct ion .
8/12/2019 MATH 37 UNIT 2.2
32/32
END