Integration By Parts-tabular Method l6l21

Integration by PartsTabular Method

Integration by Parts

udv  uv  vdu  

LIATE Rule A rule of thumb to select u. ◦ L: Logarithmic functions: ln x, log10 x, etc. ◦ I: Inverse trigonometric functions: tan-1 x, sec1 x, etc. ◦ A: Algebraic functions: x3, 3x+1, etc. ◦ T: Trigonometric functions: sin x, cos x, etc. ◦ E: Exponential functions: ex, 2x, etc.

Example 1

 x sin xdx 

Sign

  

d u dx

x 1 0

 dv sin x  cos x sin x

x sin x dx   x cos x  sin x  C 

Example 2

x

Sign

   

x

2

2

sin xdx  d u dx

x2 2x 2 0

 dv sin x  cos x sin x cos x

sin xdx   x cos x  2 x sin x 2

 2 cos x  C

Example 3

 (2 x

Sign

3

 5x)e dx  2x

d u dx



2x3 5x



6 x2  5



 

 dv e2x e 2x 2

12x

e 2x 4

12

e 2x 8

0

e 2x 16

3 2 2 x  5 x 6 x  5 2 x 12 x 2 x 12 2 x 3 2x 2x ( 2 x  5 x ) e dx  e  e  e  e C  2 4 8 16 3    e2 x  x3  x 2  4 x  2   C 2  

Example 4

x

ln xdx  d u dx

Sign

 

3

 dv

ln x

x3

1 x

x4 4



4  x 1 x  3   x ln x dx  ln x  dx      4 4 x    4 4 x x  ln x   C 4 16 4

Example 5

 ln xdx  d u dx

Sign

 

ln x 

1 x

 dv 1 x

1 ln x dx  x ln x  x dx     x   x ln x  x  C

Example 6

 tan

Sign

 



1

xdx 

d u dx tan1 x

 dv

1 1 x2

x

x  tan xdx  x tan x   1  x 2 dx 1 1  x tan x  ln 1  x 2  C 2 1

1

1 x  1  x 2 dx substitute t  1  x 2 dt  2 xdx 1 1   dt 2 t 1  ln t  C 2 1  ln 1  x 2  C 2

Example 7

e

x

e

x

sin xdx 

Sign

d u dx

 dv

  

sin x cos x sin x

ex ex ex



sin xdx  e sin x  e cos x   e sin xdx x

x

x

2 e sin xdx  e sin x  e cos x x

x

x

1 x  e sin xdx  2 e sin x  cos x   C x

Example 8

e

2x

cos xdx 

Sign

d u dx

 dv



cos x

e2 x

sin x  cos x

e2x 2 e2x 4

 



2x

2x

2x

e e e  e cos xdx  2 cos x  4 sin x   4 cos xdx 5 2x e2 x e2 x e cos xdx  cos x  sin x  4 2 4 2x e 2x  e cos xdx  5 2 cos x  sin x   C 2x