SECTION 5-2Integration of ln

Definition:

The natural logarithm is the function defined on the interval ( 0, ) where

x

dtt

x1

1ln

The “ln” stands for the Latin Logarithmus Naturalis

Recall

If y = ln(x), then

Therefore

xdx

dy 1

dxxdx

dy 1

Cxdx

dxx

y

)ln( x

1

1

Integration Rules for ln

Let u be a differentiable function in terms of x and du = u’dx

Cuduu

||ln1

Cuduu

u ln

'

1) Evaluate

72 xu

dx

x 72

5

dxx 72

15

dxdu 2

2) Evaluate

3

1210dx

x

x

210 xu

xdxdu 2

dxdux

2

1

1)3(,9)1( uu

1

9 2

1duxu

x

3) Evaluate dxx

x)ln(

)ln(xu

dxx

du1

dxxdu

xdux

u

4) Evaluate

4

1 1 xx

dx

xu 1

dxx

du2

1

dxdux 2

2)1(,3)4( uu

duxux

2)(

13

2

5) Evaluate

dxx

xx3)1(

)2(

duu

uu3

)21)(1(

1xu

dxdu xu 1

6) Evaluate

Divide using synthetic division

dxx

xx

5

2063

Assignment

Page 340 # 1-15 all, 27, 53, 54, and 56