The law of mass action results in the differential

The Law of Mass Action results in the differential equation
  
q B q A k
dt
dq
   where
x
is the
amount of substance at time
t
resulting from the reaction of two other substance and
A
,
B
are
constants.
The production of the pancreatic enzyme trypsin from trypsinogen in digestion conforms with the
Law of Mass Action. The production is modeled by
  
q B q A k
dt
dq
   where
A
is the initial
amount of trypsinogen and
B
is the initial amount of trypsin.
Solving for
q
yields
  
 

 
dt
q B q A k
dq
.

MATH 37
Lecture Guide UNIT 2 albabierra
13
Exercise Items. Try to solve the following.
Evaluate the following by decomposing the integrands to sums of partial fractions.
1.



dx
x
x
9
3 5
2
6.

  
   
dx
x x x
x x x x
3 5
17 4 5 3
2 3
2 3 4
Long division!!!
2.

 

dx
x x
x
4 3
11
2
7.
 
 

  

dx
x x x
x
2 2 2
6
2
2
3.

 
 
dx
x x x
x x
2
4 2
2 3
2
8.
  
 

  
 
dx
x x x
x x
1 3 2
1 8
2
2 3
4.


 
dx
x x
x x
3
2
1 4
9.
  



  
 
dx
x x x
x x
1 3 2
7 2 7
2
2
5.


 
dx
x x
x x
4
8 2
3
2
10.   

  

dx
x x x
x x
3 2 3
9 2
2 2
3
For the following, perform long division before decomposing the integrand to partial fractions.
11.

 
dx
x x
x
2
3
2
3
12.


 
dx
x x
x x
4
8 8
3
2 4
The following are cases of repeated factors in the denominators.
13.
 



dx
x x
x
2
2
1
2
15.
 
 

 
 
dx
x x
x x
2
2
2
2 3
22 15 6
14.
 


3 2
1 x
dx
16.

 

dx
x x x
x
2 3 4
4 4
3
(Physics) Suppose the velocity of a point moving along a coordinate line
1
3
3


t
t
ft/sec, where t is the
time in seconds. How far does the point travel during the time travel from 1  t to 2  t ?
2.
 



dx
x x
x
2
2
2
1
2
Part A. Part B.

MATH 37
Lecture Guide UNIT 2 albabierra
14
(Population studies) In many population growth models, there is an upper limit beyond which the
population cannot grow. Let us suppose that the earth will not support a population of more than 16
billion and that there were 2 billion people in 1925 and 4 billion in 1975. Then if
y
is the population
t
years after 1925, an appropriate model is the differential equation
 
y ky
dt
dy
  16 . (i.) Solve for y (a
s a function of t ). (ii.) Find the population in 2015. (iii.) When will the population be 9 billion?
__________________________
2.5 Other Substitution Techniques (TC7 pp. 614-619 / TCWAG pp. 566-569)
For rational functions of sine and cosine, use the substitute:
 
2
1
tan z 
  cos ,   sin ,   d
Exercise Items. Evaluate the following using a proper substitute.
1. 


d
cos
cos 
 5 3
5.

 

2 3 cos
d
2.

 3 5 

sin
d
6.

  

tan sin
d
3.

  

cos sin
d
3 4
7. 


d
cos
sin

 2
2
4.

   

cos sin
d
1
8. 


d
sin
tan

2
1
Must try!!!

  

cos sin
d
TO DO!!! Evaluate the following by using proper substitutes.
1.

   

2 cos sin
d
2.

  

tan sin
d

MATH 37
Lecture Guide UNIT 2 albabierra
15
For varying rational exponents or radicals, use the substitute:
n
z x  where n is the “ proper ” exponent to remove all rational exponents in
the resulting integral
Exercise Items. Evaluate the following using a proper substitutes.