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2.2

POWERS OFTRIGONOMETRIC

FUNCTIONS

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Solvable integral forms . . .

Cxcosxdxsin

Cxsindxcos Cxseclnxdxtan

Cxcsclnxdxcot Cxtanxseclnxdxsec

Cxcotxcsclnxdxcsc

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Solvable integral forms . . .

Cxtanxdxsec 2

Cxcotxdxcsc

2

Cxsecxdxtanxsec Cxcscxdxcotxcsc

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Restr ict ions

xdxsinm

xdxcos

n

xdxcosxsin nm

xdxtan

m

xdxcot

n

Form 1.

Form 2.

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Restr ict ions

Form 3.

Form 4.

xdxsec

m

xdxcsc

m

xdxsectan nm

xdxcscxcot nm

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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 .

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Form 1

xdxsinm

xdxcos

n

xdxcosxsin nm

Case 2. m and n are even.

Use2

212 xcosxsin

2

212 xcosxcos

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Example.

Evaluate .

xdxsin3

Solut ion:

xdxsin

3

dxxsinxsin

2

dxxsinxcos21

Let .xcosu dxxsindu

dxxsindu

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Solut ion (cont inued)

xdxsin3

duu21

Cuu

3

3

Cxcosxcos 3

31

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Example.

Evaluate . xdxcosxsin 32

Answer : Cxsinxsin

53

53

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Example.

Evaluate .

xdxcosxsin 22

Solut ion:

xdxcosxsin 22

dxxcosxcos 2121

4

1

dxxcos 22141

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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

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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

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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 .

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Example.

Evaluate .

xdxtan3

Solut ion:

xdxtan3

dxxtanxtan

2

dxxsecxtan 12

dxxtanxsecxtan 2

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Solut ion (cont inued)

Cxseclnxtan

2

2

dxxtanxsecxtan 2

xdxtan3

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Example.

Evaluate . xdxcot4

Answer : Cxxcotxcot

3

3

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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 .

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Use In teg rat ion by Parts w ith

xdxsecdv 2

xdxcscdv

2

Case 2. m is odd .

Form 3. xdxsecm xdxcscm

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Example.

Evaluate .

xdxcsc4

Solut ion:

dxxcscxcsc

22

dxxcscxcot 221

xdxcsc4

Let .xcotu dxxcscdu 2

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Solut ion (cont inued)

xdxcsc4

dxxcscxcot 221

duu21

Cu

u

3

3

xcot Cxcot

3

3

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Example.

Evaluate .

xdxsec3

Solut ion:

xdxsec

3

u dv dxxsec xsec2

du xdxtanxsec v xtan

By IBP!

xdxsec3 dxxsecxtanxtanxsec 2

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Solut ion (cont inued)

xdxsec

3

dxxsecxtanxtanxsec

2

dxxsecxsecxtanxsec 12

dxxsecdxxsecxtanxsec 3

Hence,

xdxsec32 dxxsecxtanxsec

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Answer :

xdxsec3

Cxtanxseclnxtanxsec

2

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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 .

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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

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Example.

Evaluate .

Solut ion:

xdxsecxtan 42

xdxsecxtan 42

dxxsecxsecxtan 222

dxxsecxtanxtan 222 1

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Solut ion (cont inued)

Let .xtanu xdxsecdu 2

duuu 22 1 xdxsecxtan 42

duuu 42

C

uu

53

53

Cxtanxtan

53

53

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Example.

Evaluate .

Answer :

xdxcscxcot 33

Cxcscxcsc

53

53

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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 .

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END