1. The Natural Logarithmic Function and Integration
5.2
The Natural Logarithmic
Function and Integration
Log Rule for Integration

2. Ex.
Let u = 2x - 1
Ex.
du = 2 dx

3. Ex. Find the area of the region bounded by the graph of
, the x-axis, and the line x = 3. .5
Let u = x2 + 1
du = 2x dx 3

4. Recognizing
Always ask yourself, is the top the
derivative of the bottom.
What does the numerator need to become the derivative of
the denominator?
It needs to be multiplied by 2.

5. Using long division before integrating.
If a rational function has
a numerator of degree
greater than or equal to
that of the denominator,
division may reveal a form
to which we can apply the
Log Rule.
Long division
gives us…

6. Change of Variables with the Log Rule
Let u = x + 1
du = dx and
x = u - 1

7. Evaluate
Divide top and bottom by x
and you’ll get a u’/u.
Day 1 stop
Integrals of 6 basic trigonometric functions.

8. Evaluate

9. The electromotive force E of a particular electrical circuit is
given by E = 3 sin 2t where E is measured in volts and t is
measured in seconds. Find the average value of E as t ranges
from 0 to .5 seconds.
Average Value is given by
Let u = 2t
du = 2 dt

10. 1