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CALCULUS: Graphical,Numerical,Algebraic by Finney,Demana,Watts and Kennedy
Chapter 6: Differential Equations 6.2: Integration by Recognition
14u 2148 )24
3
u 9
)21
2xu 2cos )18
243224
2
22
xxdxxxxx
x
x
dx
dxxx
What you'll Learn About How to integrate a product by recognizing that one of the pieces contains the
derivative of the other
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dxxA 5
2 ) dxxB 14 )
12 ) 2
dxxxC 13 )3132
dxxxD
dxtttD 84232 ) 32
25 )
2
dxx
xE
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cossin )
sin )
57cos )
4
32
xdxxF
dxxxE
dxxD
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sectan ) 2 xdxxG
dxx
x
sin34
cos )58
16
1 )
81
1 )
22 dx
xK dx
xJ
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81
2 )
2 dx
x
xI
dxx
xNdx
x
xM
dxx
xL
9
32
2
ln )
20
4 )
20
4 )
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2
0
4
3
4
3 )66
cot )64
dxe
e
xdx
x
x
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If the denominator’s quantity is to the first, the antiderivative is either
arctangent or natural log (ln)
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CALCULUS: Graphical,Numerical,Algebraic by Finney,Demana,Watts and Kennedy
Chapter 6: Differential Equations 6.3: Tabular Integration
Proof of Integration by
Parts
uvdx
dFind .1
2. Integrate both sides
3. Solve for udv
Use ultra violet minus super vdu to integrate the following
.
xxe .2
Use tabular integration to integrate the following
xxe .2
What you'll Learn About How to integrate a product by that cannot be done by recognition
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Use tabular integration to integrate the following
-x2ex .6
2
cx .8 2 xos
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Solve the initial value problem using tabular integration
0 xand 2y sin2dx
dy .11 xx
0-1)(y 22dx
dy .16 xx
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Use tabular integration to integrate the following
xdxlnx .10 2
Use ultra violet minus super vdu to integrate the following
xdxlnx .10 2
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Use tabular integration to integrate the following
dxxA arcsin .
dxxex 2cos .19
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Top Heavy Integrals
dxx
xxA
2
.
dxx
xB
5 .
dxx
xxC
2 .
3
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CALCULUS: Graphical,Numerical,Algebraic by Finney,Demana,Watts and Kennedy
Chapter 6: Differential Equations 6.5: Partial Fractions
dxxx
xA
4
12 )
2
dxxx
xB
103
16 )
2
What you'll Learn About How integrate a fraction when the denominator can be factored and the numerator
is not the derivative of the denominator
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dxxx
C 132
2 )
2
dxx
xD
1
5 )
2
3
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xx
xE
3
32(x)f )
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1
9 )
2xF
1
9 )
2x
xG
1
9 )
2x
xH
1
9 )
2xI
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CALCULUS: Graphical,Numerical,Algebraic by Finney,Demana,Watts and Kennedy
Chapter 6: Differential Equations 6.5: Logistic Growth Fractions
I am sick (Initial Value). Eventually everyone gets sick(Max). So what happens to
the rate of people getting sick. People will get sick quickly, then it will be harder to
find people that aren't sick yet (rate slows down-point of inflection) and eventually
everyone gets sick.
This is similar to a rumor spreading or facebook/twitter accounts.
yet)sick (Not peoplehealthy :P-M
peoplesick the toalproportiondirectly :kp
sick getting people ofgrowth of rate
)(
dt
dP
PMkPdt
dP
Remember directly proportional is just like P = 8.50h (Your pay is
directly proportional to the amount of money you make which can
change) That 8.50 is your k.
What you'll Learn About How to recognize a logistical growth differential equation
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In 1985 and 1987, the Michigan Department of Natural Resources
airlifted 61 moose form Algonquin Park, Ontario to Marquette County in
the Upper Peninsula. It was originally hoped that the population P would
reach carrying capacity in about 25 years with a growth rate of
)1000(0003. PPdt
dp
Solve the differential equation with the initial condition P(0) = 61.
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24. Which of the following differential equations for a population P could model
the logistic growth shown in the figure above?
0.0002P08.0dt
dP )
.000208.0dt
dP D) 0002.08.
dt
dP )
0002.08.dt
dP B) 0008.002.
dt
dP )
2
22
22
PE
PPC
PPPPA
21. The number of moose in a national park is modeled by the function M that
satisfies the logistic differential equation .05 11000
dM MM
dt
, where t is
the time in years and M(0)=50. What is the lim ( )?t
M t
A) 50 B) 200 C) 500 D) 1000 E)
2000
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84. The rate of change, ,dP
dt of the number of people on an ocean beach is
modeled by a logistic differential equation. The maximum number of people
allowed on the beach is 1000. At 10 A.M., the number of people on the
beach is 400 and is increasing at the rate of 200 people per hour. Which of
the following differential equations describes the situation.
1000200dt
dP )
10001200
1
dt
dP D) 1000
3
1
dt
dP )
10010002
1
dt
dP B) 1000
200
1
dt
dP )
PPE
PPPC
PPPA
26. The population P(t) of a species satisfies the logistic differential equation
4 ,2000
dP PP
dt
where the initial position P(0)=1500 and t is the time in years.
What is ?)(lim tPt
A) 2500 B) 8000 C) 4200 D) 2000 E) 4000
Let g be a function with g(4) = 1, such that all points (x, y) on the graph of g
satisfy the logistic differential equation 3 2 .dy
y ydx
b) Given that g(3) = 1, find lim ( ) and lim ( ).x x
g x g x
c) For what value of y does the graph of g have a point of inflection?
Find the slope of the graph of g at the point of inflection. (It is not
necessary to solve for g(x).)
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A population is modeled by a function P that satisfies the logistic differential
equation
1 .10 15
dP P P
dt
a) If P(0) = 3, what is the lim ( )?
tP t
If P(0) = 20, what is the lim ( )?t
P t
b) If P(0) = 3, for what value of P is the population growing the fastest?
Rogawski
8. If y(x) is a solution to 3 (10 )dy
y ydx
with y(0) = 3 then as x,
A) y(x) increases to
B) y(x) increases to 5
C) y(x) decreases to 5
D) y(x) increases to 10
E) y(x) decreases to 10
9. If y(x) is a solution to 4 (12 )dy
y ydx
with y(0) = 10 then as x,
A) y(x) decreases to
B) y(x) increases to 6
C) y(x) increases to 12
D) y(x) decreases to 12
E) y(x) decreases to 0
16. If 3 (10 2 )dy
y ydt
with y(0) = 1 then, y is increasing the fastest when
A) y = 1.5
B) y = 2.5
C) y = 3
D) y = 4
E) y = 5
18. If 3 (10 2 )dy
y ydt
with y(0) = 1, then the maximum value of y is
A) y = 1
B) y = 2.5
C) y = 5
D) y = 10
E) Never attained; has no maximum value
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Princeton Review (p. 806)
25. Given the differential equation 650
dz zz
dt
, where z(0) = 50, what is the
?)(lim tzt
A) 50 B) 100 C) 300 D) 6 E)
200
25. Given the differential equation 650
dz zz
dt
, where z(0) = 50, then z is
increasing the fastest when z =
A) 150 B) 100 C) 300 D) 50 E) 100
Other Rate type problems
Rogawski
11. The rate at which a certain disease spreads is proportional to the quotient of
the percentage of the population with the disease and the percentage of the population
that does not have the disease. If the constant of proportionality is .03, and y is the
percent of people with the disease, then which of the following equations gives R(t),
the rate at which the disease is spreading.
) R(t) .03y
.03B) R(t)
dr .03)
dt (1 )
yD) R(t) .03
(1 - y)
dr) .03
dt
A
dy
dt
RC
R
E R
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12. The rate of change of the volume, V, of water in a tank with respect to time, t
is directly proportional to the square root of the time, t, it takes to empty the tank .
Which of the following is a differential equation that describes this relationship.
dV) ( ) ) ( ) )
dt
dV dV) )
dt dt
A V t k t V t k V C k t
kD E k V
V
B
16. Let P(t) represent the number of wolves in a population at time t years, when
0t . The population P(t) is increasing at rate directly proportional to 500 divided
by P(t), where the constant of proportionality is k. Write the differential equation
that describes this relationship.
23. If P(t) is the size of a population at time t, which of the following differential
equations describes exponential growth in the size of the population.
2
2
dP dP dP) 200 ) 200 ) 100
dP dP) 200 ) 100
A B t C tdt dt dt
D P E Pdt dt
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8.4 Improper Integrals
1 31 )2
x
dx
1 3
2 )6
x
dx
What you'll Learn About How to integrate functions that approach infinity or functions that approach an
asymptote
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0
32
)10x
dx
0
2 34
2 )14
xx
dx
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0
2 )18 dxex x
43. Find the area of the region in the first quadrant that lies under the
given curve
2
ln
x
xy
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dxxe x2
2 )22
1
0 21 )26
x
dx
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4
1 )30
x
dx
2
0 1 )41
x
dx
