The Natural Log Function: Integration

Lesson 5.7

Log Rule for Integration

• Because

Then we know that

• And in general, when u is a differentiable function in x:

1ln( )

dx

dx x

1lndx x C

x

1lndu u C

u

Try It Out

• Consider these . . .

2

33

xdx

x

2sec

tan

xdx

x

Finding Area

• Given

• Determine the area under the curve on the interval [2, 4]

2

lny

x x

Using Long Division Before Integrating

• Use of the log rule is often in disguised form

• Do the division on this integrand and alter it's appearance

22 7 3

2

x xdx

x

2

2 11 Remainder 19

2 2 7 3

x

x x x

Using Long Division Before Integrating

• Calculator also can be used

• Now take the integral

192 11

2x dx

x

Change of Variables

• Consider

Then u = x – 1 and du = dx But x = u + 1 and x – 2 = u – 1

• So we have

Finish the integration

3

2

1

x xdx

x

3

1 1u udu

u

Integrals of Trig Functions

• Note the table of integrals, pg 357

• Use these to do integrals involving trig functions

tan 5 d

1

0

sin cost t dt

Assignment

• Assignment 5.7

• Page 358

• Exercises 1 – 37 odd 69, 71, 73