5.2 The Natural Logarithmic Function: Integration

Log Rule for Integration – Let u be a differentiable function of x.

1. 1

lndx x Cx

2. 1

lndu u Cu

Examples: Find the indefinite integral.

1. 10dx

x

2. 1

5dx

x

3. 1

4 3dxx

4. 2

35

xdx

x

5. 2

3 2

2

3

x xdx

x x

6. 3

1

lndx

x x

7. 3 24 7 9

3

x x xdx

x

Guidelines for Integration:

1. Learn a basic list of integration formulas. Now you know about 12, soon you will know 20.

2. Find an integration formula that resembles all or part of the integrand, and, by trial and error,

find a choice of u that will make the integrand conform to the formula.

3. If you cannot find a u-substitution that works, try altering the integrand. You might try a trig

identity, multiplication and division by the same quantity, addition and subtraction of the same

quantity, or long division. Be creative.

4. If you have access to computer software that will find anti-derivatives symbolically, use it.

Integrals of the Six Basic Trig Functions –

sin cos cos sin

tan ln cos cot ln sin

sec ln sec tan csc ln csc cot

udu u C udu u C

udu u C udu u C

udu u u C udu u u C

Examples: Find the indefinite integral.

1. tan5 d

2. sec2

xdx

3. 2csc

cot

tdtt

Examples: Solve the differential equations. Find the equation that passes through the given point.

1. 2, 1,0

dy x

dx x

2. 2sec, ,4

tan 1

dr t

dt t

Examples: Evaluate the definite integral.

1. 1

1

1

2 3dx

x

2. 2 1

ln

e

edx

x x

3. 2

1

1 cos

sind