Integration by Parts-Tabular Method

Integration by Parts

vduuvudv

LIATE Rule

A rule of thumb to select u.◦L: Logarithmic functions: ln x, log10 x, etc.

◦I: Inverse trigonometric functions: tan-1 x, sec-1 x, etc.

◦A: Algebraic functions: x3, 3x+1, etc. ◦T: Trigonometric functions: sin x, cos x,

etc. ◦E: Exponential functions: ex, 2x, etc.

dxxx sin

Sign udx

d dvx

10

xsinxcosxsin

Cxxxdxxx sincossin

Example 1

dxxx sin2

Sign udx

d dv2xx2

2

xsinxcosxsin

Cx

xxxxdxxx

cos2

sin2cossin 22

0 xcos

Example 2

dxexx x23 )52(

Sign udx

d dvxx 52 3

56 2x

x12

xe2

2

2xe

Cxxxe

Ceex

ex

exx

dxexx

x

xxxxx

242

3

16

12

8

12

4

56

2

52)52(

232

2222

23

23

12

0

4

2xe

8

2xe

16

2xe

Example 3

dxxx ln3

Sign udx

d dvxln

x

1

3x

4

4x

Cx

xx

dxx

xx

xdxxx

16ln

4

4

1ln

4ln

44

443

Example 4

dxxln

Sign udx

d dvxln

x

1

1x

Cxxx

dxxx

xxdxx

ln

1lnln

Example 5

dxx1tan

Sign udx

d dvx1tan

21

1

x

1x

Cxxx

dxx

xxxdxx

21

211

1ln2

1tan

1tantan

Cx

Ct

dtt

xdxdt

x

dxx

x

2

2

2

1ln2

1

ln2

1

1

2

1

2

1 tsubstitute

1

Example 6

dxxex sin

Sign udx

d dvxsinxcosxsin

dxxexexedxxe xxxx sincossinsin

xexexe

Example 7

xexedxxe xxx cossinsin2 Cxxedxxe xx cossin

2

1sin

dxxe x cos2

Sign udx

d dvxcos

xsinxcos

dxxe

xe

xe

dxxexxx

x cos4

sin4

cos2

cos222

2

xe2

2

2xe

Example 8

4

2xe

xe

xe

dxxexx

x sin4

cos2

cos4

5 222

Cxxe

dxxex

x sincos25

cos2

2