or keyword: use double angle formulas

dxxSin 2

dxSinx dxCosx dxTanx dxCotx dxSecx dxCscx

dxxCos 2 dxxTan 2 dxxCot 2 dxxSec 2 dxxCsc 2

dxxSin3 dxxCos 3 dxxTan 3 dxxCot 3 dxxSec3 dxxCsc 3

dxxSin 4 dxxCos 4 dxxTan 4 dxxCot 4 dxxSec 4 dxxCsc 4

dxxSin5 dxxCos 5 dxxTan 5 dxxCot 5 dxxSec5 dxxCsc 5

CxCosdxSinx

dxSinx

CxSindxCosx

dxxCos

CxCosdxTanx ||ln

CxSecdxxTan ||ln

or

dxxTan

CxSindxxCot ||ln

dxxCot

CxTanxSecdxxSec ||ln

dxxSec

CxCotxCscdxxCsc ||ln

CxCotxCscdxxCsc ||ln

or

dxxCsc

dxxSin 2

Keyword: Use Double Angle Formulas

CxSinxdxxCosdxxCosdxxSin

22

21)21(

21

2)21(2

dxxCos 2

Keyword: Use Double Angle Formulas

CxSinxdxxCosdxxCosdxxCos

22

21)21(

21

2)21(2

dxxTan 2

Keyword: Use Pythagorean Identities

CxxTandxxSecdxxTan )1( 22

dxxCot 2

Keyword: Use Pythagorean Identities

CxxCscdxxCscdxxCot )1( 22

dxxSec 2

CxTandxxSec 2

dxxCsc 2

CxCotdxxCsc 2

dxxSin3

CxCosCosxdxSinxxCosdxSinx 32

31.

Keywords: Break into a single sine term and the left-over terms; Then use a Pythagorean Identity on the left-over terms.

dxSinxxCosdxSinxxSindxxSin )1(. 223

dxxCos 3

CxSinSinxdxCosxxSindxCosx 32

31.

Keywords: Break into a single cosine term and the left-over terms; Then use a Pythagorean Identity on the left-over terms.

dxCosxxSindxCosxxCosdxxCos )1(. 223

dxxTan 3

CCosxxTandxTanxdxxSecTanx ||ln21. 22

Keywords: Break into a tangent squared term and the left-over terms; Then use a Pythagorean Identity on the tangent squared term.

dxxSecxTandxxTanTanxdxxTan )1(. 223

dxxCot 3

CSinxxCotdxCotxdxxCscCotx ||ln21. 22

Keywords: Break into a cotangent squared term and the left-over terms; Then use a Pythagorean Identity on the cotangent squared term.

dxxCscxCotdxxCotCotxdxxCot )1(. 223

dxxSec3

dxxSecSecxdxxSecI 23 .dxTanxSecxdu

Secxu

Keywords: Special use of Integration by Parts

xTanvdxxSecdv

2

dxxSecSecxTanxSecxdxxTanSecxTanxSecx )1( 22

dxSecxdxxSecTanxSecx 3

||ln TanxSecxITanxSecxI

||ln2 TanxSecxTanxSecxI

CTanxSecxTanxSecxI ||ln21

21

vduuv

dxxCsc 3

dxxCscCscxdxxCscI 23 .dxCotxCscxdu

Cscxu

Keywords: Special use of Integration by Parts

CotxvdxxCscdv

2

dxxCscCscxCotxCscxdxxCotCscxCotxCscx )1( 22

dxCscxdxxCscCotxCscx 3

||ln CotxCscxICotxCscxI

||ln2 CotxCscxCotxCsxxI

CCotxCscxCotxCscxI ||ln21

21

vduuv

dxxSin 4

dxxCosxCosdxxCosdxxSin )2221(41

221 2

24

dxxCosxCosdxxCosxCos 42122

23

41

2)41(221

41

Keywords: Use Double-Angle Formulas twice

CxSinxSinx

4

812

23

41

dxxCos 4

dxxCosxCosdxxCosdxxCos )2221(41

221 2

24

dxxCosxCosdxxCosxCos 42122

23

41

2)41(221

41

Keywords: Use Double-Angle Formulas twice

CxSinxSinx

4

812

23

41

dxxTan 4

dxxSecxTandxxTandxxSecxTan )1(31 23222

dxxSecxTandxxTanxTandxxTan )1(. 22224

Keywords: Break in to a tangent squared term and the left-over terms. Use Pythagorean Identity on the tangent squared term

CxTanxxTan 3

31

dxxCot 4

dxxCscxCotdxxCotdxxCscxCot )1(31 23222

dxxCscxCotdxxCotxCotdxxCot )1(. 22224

Keywords: Break in to a cotangent squared term and the left-over terms. Use Pythagorean Identity on the cotangent squared term.

CxCotxxCot 3

31

dxxSec 4

CxTanTanxdxxSecxTandxxSec 3222

31

xdxSecxTandxxSecxSecdxxSec 22224 )1(.

Keywords: Break in to a secant squared term and the left-over terms. Use Pythagorean Identity on the left-over terms.

dxxCsc 4

CxCotCotxdxxCscxCotdxxCsc 3222

31

xdxCscxCotdxxCscxCscdxxCsc 22224 )1(.

Keywords: Break in to a cosecant squared term and the left-over terms. Use Pythagorean Identity on the left-over terms.

dxxSin5

dxSinxxCosdxSinxxSindxxSin 2245 )1(.

dxSinxxCosdxSinxxCosdxSinxdxSinxCosxCos 4242 2)21(

Keywords: Break into a single sine term and the left-over terms; Then use Pythagorean Identity on the left-over terms.

CxCosxCosCosx 53

51

32

dxxCos 5

dxCosxxSindxCosxxCosdxxCos 2245 )1(.

dxCosxxSindxCosxxSindxCosdxCosxSinxSin 4242 2)21(

Keywords: Break into a single cosine term and the left-over terms; Then use Pythagorean Identity on the left-over terms.

CxSinxSinSinx 53

51

32

dxxTan 5

dxxSecxTandxTanxTandxxTan )1(. 23235

dxxTanTanxxTandxxTandxxSecxTan 24323 .41

Keywords: Break into a tangent squared term and the left-over terms; Then use Pythagorean Identity on the tangent squared terms.

dxTanxdxxSecTanxxTandxxSecTanxxTan 2424

41)1(

41

CCosxxTanxTan ||ln21

41 24

dxxCot 5

dxxCscxCotdxCotxCotdxxCot )1(. 23235

dxxCotCotxxCotdxxCotdxxCscxCot 24323 .41

Keywords: Break into a cotangent squared term and the left-over terms; Then use Pythagorean Identity on the cotangent squared term.

dxCotdxxCscxCotxCotdxxCscCotxxCot 2424

41)1(

41

CSinxxCotxCot ||ln21

41 24

dxxSec5

Keywords: Special use of Integration by Parts

dxxCsc5

Keywords: Special use of Integration by Parts

CosxSinxdxd

)( SinxCosxdxd

)(

xSecTanxdxd 2)( xCscCotx

dxd 2)(

TanxSecxSecxdxd

)( CotxCscxCscxdxd

)(

122 CosSin

22 1 TanSec

22 1 CotCsc

222 SinCosCos

122 2 CosCos

2212 SinCos

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