TECHNIQUES OF INTEGRATION TECHNIQUES OF INTEGRATION In this chapter we develop techniques for using...

TECHNIQUES OFINTEGRATION

In this chapter we develop techniques for using these basic integration formulas to obtain indefinite integrals of more complicated functions.

INTEGRATION BY PARTS

formula for integration by parts.

duvvudvu

Example

Find dxex x

Example

Find dxxx sin

xu dxedv x

dxdu xev xu xdxdv sin

dxdu xv cos

dxexedxex xxx


formula for integration by parts. duvvudvu

Example

Find

xu ln dxdv

dxdux

1 xv dxx)ln(

REMARK1: aim in using integration by parts is to obtain a simpler integral than the one we started with.

REMARK2: How to choose u and dv to obtain simpler integral


Example

Find dtet t 2

2t te

2

t2

0

tetete

diff

REMARK2: in some integral, we may need to apply integration by parts many times.

2tu dtedv t

tdtdu 2 tev

duvvudvu

dtteetdtet ttt 2 22

tu 2 dtedv t

dtdu 2tev

dtetedtte ttt 222

Ceteetdtet tttt )22( 22


Example

Find dtet t 2

2t te

2

t2

0

tetete

diff

REMARK2: in some integral, we may need to apply integration by parts many times.

Example

Find dxex x 23 Example

Find dxxx sin2


formula for integration by parts.

duvvudvu

u dv

du v

diff

Example

Find dxxex sin

xsin xe

xsin

xcos xexe

diff

REMARK3: sometimes a repeated application of integration by parts leads back to an integral similar to our original one. If so, this expression can be combined with original integral.


Exam2Term082

Exam2Term102

Exam2Term092


Observe:

dxx 3)(ln

REMARK3: sometimes The reduction formula is useful because by using it repeatedly we could eventually express our integral.

partsby

dxx 2)(ln dxx)(lnpartsby

Reduction Formula

dxxnxxdxx nnn 1)(ln)(ln)(ln


Reduction Formula

)1( tan1

tantan 2

1

nxdxn

xxdx n

nn

Example dxx tan5

135

Example dxx tan6

0246


Reduction Formula

xdxxxn

xdx nn

nnn 211 cossincos1

cos

Example dxx cos5

135

Example dxx cos6

0246

Reduction Formula

xdxxxn

xdx nn

nnn 211 sinsincos1

sin

Reduction Formula

xdxncos

xdxnsin

xdxntan

xdxnsec

dxex xn

dxx n)(ln


Integration by Parts

Repeated Applications

Term 0 (diff)tabular integration

Reduction Formula

Back to original

Most of the time ln(x) is easier by

partsx

x1

1

sin

tan

1...u

TECHNIQUES OF INTEGRATION TECHNIQUES OF INTEGRATION In this chapter we develop techniques for using...

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