Transcript Basic Logarithmic and Exponential Integrals

Basic Logarithmic and
Exponential Integrals
Lesson 9.2
Review
• Recall the exception for the general power
formula
1 n 1
n
 u du  n  1 u  c n  1
• Recall also from chapter 8 that
d
1
ln u  du
dx
u
• We will use this and the fact that the
integral is the inverse operation of the
derivative
2
Filling in the Gap
• Since

d
1
ln u  du
dx
u
then
1
 u du  ln u  C
Note the absolute value requirement since we
cannot take ln u for u < 0
• Thus we now have a
way to take the integral of
when n = -1
u
du

n
3
Try It Out!
• Consider
dx
 9x  5
• What is the u?
u  9x  5
• What is the du?
du  9dx
• Rewrite, integrate, un-substitute
1 du 1
1
 ln u  C  ln 9 x  5  C

9 u 9
9
4
Integrating ex
d u
u du
• Recall derivative of exponential e  e
dx
dx
• Again, use this to determine integral
e
du

e

C

u
u
• For bases other than e
u
a
u
u
a
du

a
log a e  C 
C

ln a
5
Practice
• Try this one
 sec
2
xe
tan x
dx
• What is the u, the du?
u  tan x
du  sec x dx
2
• Rewrite, integrate, un-substitute
 e du  e
u
u
C  e
tan x
C
6
Area under the Curve
• What is the area
bounded by y = 0,
x = 0, y = e –x,
and x = 4 ?
4
e
x
dx
0
• What about volume of region rotated about
either x-axis or y-axis?
7
Application
• If x mg of a drug is given, the rate of
change in a person's temp in
4
°F with respect to dosage is T '( x) 
2x  9
• A dosage of 1 mg raises the
temp 2.4°F.

What is the function that gives total change in
body temperature?
• We are given T'(x), we seek T(x)
8
Application
• Take the indefinite integral of the T'(x)
+C
• Use the fact of the specified dosage and
temp change to determine the value of C
T ( x)  2 ln  2 x  9   2.396
9
Assignment
• Lesson 9.2
• Page 362
• Exercises 1 – 33 odd
10