𝑖𝝿
The -17𝑒 Annual Integration Games
of Improper Integrals, Integration By
Parts, and Partial Fractions
Written by Zachary Trego and Sarah
Hyman
http://prawnandquartered.com/tag/contest/
1
We hope you find this book helpful in
learning these difficult forms of
integration, and we hope you enjoy
our integration of the Hunger Games
(Pun Intended)
2
Table of Contents
• General Overviews
Integration by Parts
.……………………………………………………………………p 4
Partial Fractions
…...…………………………………………………………………… p 5
Improper Integrals ………………………………………………………………………..p 6
• Graphing calculator examples ………………………………………………………………….p 7
• Analytical Examples
Integration by Parts ………………………………………………………………………p 8,9
Partial Fractions ……………………………………………………………………………p 10,11
Improper Integrals ………………………………………………………………………..p 12
• AP Conceptual Example ……………………………………………………………………………p 13,14
• A Nice Break …………………………………………………………………………………………….p 15-21
• AP Multiple Choice Example ……………………………………………………………………..p 22-24
• AP Free Response …………………………………………………………………………………….p 25-27
• Real World Applicability ……………………………………………………………………………p 28
• Contributing Mathematician ……………………………………………………………………..p 29
• Analytical Examples …………………………………………………………………………………… P 30-33
• AP Multiple Choice ……………………………………………………………………………………p 34-38
• AP Free Response ……………………………………………………………………………….......p 39
• Works Cited …………………………………………………………………………………………….. P 40
3
Integration by Parts Overview
𝑏
𝑏
= 𝑝𝐾 −
𝑃𝑑𝐾
𝐾𝑑𝑃
𝑎
𝑎
This process should be utilized when integrating a function that contains a cyclical
5
portion. For example, in 0 x𝑒 𝑥 𝑑𝑥, the cyclical portion is 𝑒 𝑥 . This is an important skill to
have, because these integrals cannot be evaluated using normal means, including u
substitution. This should only be used for special cases that lend themselves to
integration by parts. This process can be used to integrate other functions that may not
lend themselves to integration by parts, but this is usually not necessary.
Note. It may be helpful to apply Hunger Games to these problems. Let Katniss be K and
Peeta be P as they are both characters with their own paths just as P and K are separate
functions with their own properties.
K
P
http://www.wallpaperpin.com/wallpaper/1680x1050/hunger-games-movie-wp-katniss-and-peeta-19576.html
4
Partial Fractions Overview
𝑏
Evaluate the integral :
𝑎
1
𝑑𝑥
𝑥2 − 𝑥
1. When u substitution does not work, factor
the denominator.
2. Then, separate the fraction, using different
variables for each numerator. The two
variables represent the constants that will
make the two fractions equal to the original.
3. Finally, evaluate for each variable by
substituting numerical values for x. Since x is
just a variable, not a constant, we can
substitute any value in for it.
4. Note that the values of P and K are equal in
magnitude but opposite in sign. Though this
happens from time to time, the values do not
always follow such a pattern. Always do the
work to determine these values!!!!!!
5
1
𝑥(𝑥 − 1)
=
𝑃
𝑥
+
𝐾
𝑥−1
1 = (x-1)P +(x)K
Let x=0 1= -1P
P= -1
Let x=1 1=1K
K=1
𝑏
𝑎
−1
1
+
𝑑𝑥
𝑥
𝑥−1
Improper Integrals Overview
Whenever you see an integral with one or more undefined endpoints, rewrite the
𝑎
integral using limits before evaluating. ∞
𝑓 𝑥 𝑑𝑥 = lim 𝑓 𝑥 𝑑𝑥
𝑎→∞
0
0
As the function approaches infinity, the speed at which it approaches a certain
value determines whether or not the integral exists or not.
These integrals can either be determinate or indeterminate. An indeterminate
integral would contain an undefined value or approach either -∞ or ∞.
𝑎
* For integrals that resemble the form 𝑥 𝑝 , the integral converges for every value
such that p > 1. This property reflects the speed at which the function approaches
0.
51
51
𝑑𝑥
=
lim
dx
0
+ 𝑎
𝑥
𝑎→0
10
Look out for these integrals too!
0
𝑥
1
𝑑𝑥 = lim−
𝑎→3
𝑥−3
6
𝑎
0
1
𝑑𝑥 + lim+
𝑏→3
𝑥−3
10
𝑏
1
𝑑𝑥
𝑥−3
Graphing Calculator Examples for Determinate and
Indeterminate Integrals of Functions
Indeterminate
1
𝑦=
𝑥
𝑦=
Determinate
1
𝑦= 2
𝑥
1
𝑥
𝑦=
1
𝑥3
Notice the speed at which the functions approach the x-axis. The fuctions 𝑦 =
𝑦=
1
𝑥3
clearly approach the x-axis much faster than the functions y=
functions y=
1
𝑥
1
1
𝑥
1
𝑥2
and
1
𝑥
𝑎𝑛𝑑 𝑦 = . The
and y=𝑥 are not integrable as x approaches infinity. Thus, if the degree
of x in the denominator is greater than 1, and the numerator contains just a constant,
then the improper integral is determinate.7
Analytical Example
24
Integrate the following integral
𝑥𝑒 𝑥 𝑑𝑥
0
http://thehungergames.wikia.com/wiki/74th_Hunger_Games
8
Solution
24
1. First, using the general form of
integration by parts, determine
what parts of the function to
designate as P and dK. Assign the
part that will become a constant
when differentiated as P. Assign
the easily integrable part as dK. In
this example, x will be P and 𝑒 𝑥 𝑑𝑥
will be dK.
2. Then, take the derivative of P, and
take the integral of dK.
3. Lastly, use P, K and dP in the
general form of integration by
parts to evaluate the integral.
𝑥𝑒 𝑥 𝑑𝑥
0
P=x
dK= 𝑒 𝑥 𝑑𝑥
dP=1 dx
K= 𝑒 𝑥
24
𝑥 𝑑𝑥 = x 𝑒 𝑥
𝑥𝑒
0
24 𝑥
𝑒 1𝑑𝑥
0
( xe e )
x
x
-

24
0
24e 24  e 24  (0  1)
9
Analytical Example
Throughout the Hunger Games, Katniss and Peeta are forced to
separate in order to survive, but they eventually reunite to
achieve victory. This parallels the process by which you solve
partial fractions, because you must separate the function in order
to achieve integration victory. Also, you can plug in any value for
x in order to solve for K and P, because the environment, which is
variable, in which the Hunger Games takes place can be anything
from a desert to a tundra. If Katniss and Peeta are in the Integral
Games for 18.358 days, and their journeys are modeled by the
5𝑥
function f(x)=14𝑥2 −6𝑥−44 . What is the total displacement they
traveled during the time that they were there?
5𝑥
Obviously, the function factors into (7𝑥+11)(2𝑥−4) .
10
Solution
Once the denominator is factored,
separate the fraction into two
pieces. Then, multiply both sides of
the equality by the original
denominator. Figure out which x
values will have each term, P and K,
go to zero, because having one term
go to zero allows one to easily find
the value of the other constant. Plug
in the values for P and K into the
original two separated fractions,
and integrate each fraction to
eventually solve for the original
integral.
18.358
5𝑥
𝑑𝑥
0
14𝑥 2 −6𝑥−44
𝑃
= 7𝑥+11 +
𝐾
2𝑥−4
5𝑥 = 2x − 4 P + 7x + 11 K
2
Let x=2 10= 25K K=5 Let x=
18.358
5𝑥
𝑑𝑥
0
14𝑥 2 −6𝑥−44
=
11
=
−50
𝑃
7
11
P=10
2
18.358 11 10
5
(
+
)dx
0
7𝑥+11
2𝑥−4
18.358

11  1 
  ln(7 x 11)
10  7 
0
−11 −55
7
7
18.358

2 1 

  ln(2 x  4)
5 2 
0
Analytical Example
Evaluate the integral
4 1
𝑑𝑥
0 𝑥−3
Solution
First, determine whether there are any
undefined values within the integral.
At x=3, the integral is undefined. Then,
separate the integral into two in order
to evaluate the integral from 0 to 3
and from 3 to 4. You can do this,
because there is no area at a single
point. Thus, separating the integral
accounts for the undefined value at
x=3 and does not alter the area under
the curve. If either one of the separate
integrals is indeterminate, then the
12
entire integral is indeterminate.
3 1
𝑑𝑥
0 𝑥−3
+
4 1
𝑑𝑥
3 𝑥−3
ln (x-3) + ln (x-3)
ln x 3   ln x 3 
3
4
0
3
AP conceptual example
x
4
6
8
10
12
F(x)
1
12
1
32
1
60
1
96
1
140
14
1
192
16
1
252
The function F(x) is continuous, differentiable, and constantly decreasing.
∞
Based on the values in the chart, determine whether or not 4 𝐹 𝑥 𝑑𝑥 is
determinate or indeterminate. Justify your answer.
13
Solution
𝐹 6 −𝐹(4)
6−4
*****
𝐹 8 −𝐹 6
8−6
Show that
>
or the equivalent using other points.
Since both the values and the slope of the secant lines
between values are decreasing, one can determine that the
values of the function are approaching 0 fast enough for the
∞
function to converge. Thus, 4 𝐹 𝑥 𝑑𝑥 is determinate.
14
15
http://missjordennesclass.wordpress.com/2013/03/20/what-do-you-think-of-thispuppy-its-called-a-pomsky-pomeranian-and-husky/
http://imgur.com/od0u7km
http://www.rarely-pins.com/tag/husky/
http://weheartit.com/entry/25629224
16
17
http://www.telegraph.co.uk/earth/earthpicturegalleries/8280986/Polar-bear-pictures.html?image=1
http://cutestuff.co/2011/08/newborn-tiger-cubs/
18
http://coolpets4u.blogspot.com/2012/04/kittens-and-puppies-pictures.html
19
http://www.multyshades.com/2012/06/40-heartwarming-examples-of-baby-animal-photography/
20
21
AP Level Multiple Choice Example
If
𝑑𝑦
𝑑𝑥
3𝑥
2𝑥 2 +6𝑥−8
=
Yes, you have to simplify. Why, you
ask? We want to see you suffer.
, f(x)=
3
(A)
𝑙𝑛
(2𝑥−2)10
6
(𝑥+4)5
+𝐶
3
10
6
5
(B)
𝑙𝑛 (2𝑥 − 2) (𝑥 + 4)
(C)
6
𝑙𝑛
5
(2𝑥 − 2)(𝑥 + 4) + 𝐶
(D)
3
𝑙𝑛
4
2𝑥 2 + 6𝑥 − 8 + 𝐶
(E)
+ 𝐶
3 2
x
2
2 3
x  3x 2  8 x
3
22
http://thehungergames.wikia.com/wiki/Peeta_Mellark
Answer and Solution
𝑑𝑦
3𝑥
=
𝑑𝑥
2𝑥 2 + 6𝑥 − 8
3𝑥
𝑃
𝐾
=
+
(2𝑥 − 2)(𝑥 + 4)
(2𝑥 − 2) (𝑥 + 4)
3𝑥 = 𝑃 𝑥 + 4 + 𝐾(2𝑥 − 2)
3(−4) = 𝑃 −4 + 4 + 𝐾(2(−4) − 2)
3(1) = 𝑃 (1) + 4 + 𝐾(2(1) − 2)
P=
3
5
K=
𝑦=
𝑦=
3
𝑙𝑛
10
2𝑥 − 2 +
3𝑥
3
𝑑𝑥
=
2𝑥 2 + 6𝑥 − 8
5
6
𝑙𝑛
5
𝑥+4
→
1
6
𝑑𝑥 +
2𝑥 − 2
5
1
𝑑𝑥
𝑥+4
3
10
𝑦 = 𝑙𝑛 (2𝑥 − 2) (𝑥 + 4)
Therefore, the answer to this problem is B
23
6
5
6
5
Why You Were Wrong
Choice A: You subtracted instead of added. Before simplifying, you got
ln (2 x  2)
3
10
 ln ( x  4)
6
5
+c
Be careful with sign changes.
3
Choice C: You made a mistake with the chain rule. When integrating 5
1
𝑑𝑥
2𝑥−2
6
you multiplied by two instead of dividing by two. You got 5 ln (2 x  2) + c
6
This is why you thought you could factor out the 5 .
Choice D: You made a number of errors. You thought you could use u substitution,
so you made u=2𝑥 2 +6x+8. In this case, du= 4x+ 6, but
you forgot the +6. Thus, you solved
3
4
1
𝑑𝑢
𝑢
.
Choice E: If you got this answer, we would suggest retaking AP Calculus AB. This is
just as stupid as leaving a bag of apples hanging above land mines near
your supplies, like the player from District 3.
24
AP Level Free Response (Calculator)
While Katniss Everdeen is traveling through the woods, a fire ball explodes next to her.
Consequently, she is injured and cannot walk. The fire ball generates a forest fire that is
spreading toward Katniss at a rate, in meters per minute, modeled by the function 𝑦 =
𝑒 𝑡 𝑠𝑖𝑛𝑡, where t is time in minutes. Katniss is able to limp at a rate of 6 meters per minute
toward a river near by. If the fire reaches the river in 𝜋 minutes, will Katniss reach the
river in time?
To practice integration by parts, only use a calculator
to find the actual answers after integrating.
25
http://mockingjay.net/2012/04/06/new-hq-still-katniss-running-through-fire/
Solution
𝜋 𝑥
𝑒 𝑠𝑖𝑛𝑥dx
0
𝑃 = 𝑒 𝑥 𝑑𝐾 = 𝑠𝑖𝑛𝑥𝑑𝑥
𝑑𝑃 = 𝑒 𝑥 𝑑𝑥 𝐾 = −𝑐𝑜𝑠𝑥
𝜋
𝜋
𝑥
𝑥
𝑒 𝑠𝑖𝑛𝑥𝑑𝑥 = −𝑒 𝑐𝑜𝑠𝑥 −
0
𝑃 = 𝑒 𝑥 𝑑𝐾 = 𝑐𝑜𝑠𝑥𝑑𝑥
𝑑𝑃 = 𝑒 𝑥 𝑑𝑥 𝐾 = 𝑠𝑖𝑛𝑥
𝑥
−𝑐𝑜𝑠𝑥𝑒 𝑑𝑥
0
𝜋
𝜋
𝑒 𝑥 𝑠𝑖𝑛𝑥𝑑𝑥 = −𝑒 𝑥 𝑐𝑜𝑠𝑥 + 𝑒 𝑥 𝑠𝑖𝑛𝑥 −
0
𝑒 𝑥 𝑠𝑖𝑛𝑥𝑑𝑥
0
𝜋
2
0
−𝑒 𝜋
𝑒 𝑥 𝑠𝑖𝑛𝑥
𝑑𝑥 =


(-e cos x  e sin x) 0
x
x
cos 𝜋 + 𝑒 𝜋 sin 𝜋 + 𝑒 0 cos 0 − 𝑒 𝑜 sin(0)
𝑒 𝜋 +1
2
= 12.070 meters
26
Solution Continued…
𝜋
6 dx
0
=
6 x0

= 6𝜋 − 6 0 = 18.850 meters
Katniss traveled 18.850 meters, while the fire traveled 12.070 meters in the
same amount of time. Thus, Katniss outran the fire to reach the river some time
t before time t=𝜋 minutes.
Be careful when solving this type of problem. Since it involves integration by
parts, there are a lot of steps involved. If this appears on the AP test, be sure to
show every calculus step that you took to get your answer.
27
Real World Applicability
Improper Integrals commonly pop up when dealing with probability. Integrating a
function to infinity can model the probability of an event as the probability
approaches 100%.
Integration by parts is an important tool in the field of engineering. Integration by
parts is needed in common problems, including electric circuits, heat transfer,
vibrations, structures, fluid mechanics, transport modeling, air pollution, and
electromagnetics.
Although partial fractions do not have a specific applicability, they are useful in
calculating numerous integrals. The ability to integrate more functions allows one to
be able to solve more calculus problems.
All three of these integration skills have virtually endless applications in the real
world. Integration is involved in, but not limited to, finding the area bounded by
curves, finding the volume of solids of revolution, finding the center of mass, finding
moments of inertia, calculating work done by a variable force, and finding average
values.
28
Archimedes
Archimedes, known as “the wise one,” “the master,” and “the great geometer,” was
born in 287 B.C. in the port of Syracuse, Sicily. According to ancient Greek
biographer Plutarch, Archimedes achieved so much fame because of his relation to
King Hiero II and Gelon (son of King Hiero II). He was a close friend of Gelon and
helped Hiero solve complex problem with extreme ease, utterly amazing his friend.
His greatest accomplishments were in his utilization of integration. Archimedes was
able to calculate areas under curves and volumes of certain solids by a method of
approximation, called the method of exhaustion, based on using known areas and
volumes of rectangles, discs, etc. His results were usually expressed, not in
absolute terms, but in terms of comparisons of volumes. For instance, he could
describe shapes by saying that there is a sphere of radius r surrounded exactly by a
circular cylinder of radius r and height 2r. Then, Archimedes showed that the
volume of the sphere is two thirds that of the cylinder. Archimedes found the sum
of a geometric series in such a way as to indicate that he understood the concept
of limits, which relates to improper integrals. He was also a thoroughly practical
man who invented a wide variety of machines, including pulleys and the
Archimidean screw pumping device.
29
Note: The problems are color-coded based on their difficulties. Yellow problems are
easy. Blue problems are medium. Red problem are difficult.
30
Analytical Examples
Evaluate the following integrals using integration by parts. A calculator is not required to
solve these problems.
1
1)
3𝑙𝑛10𝑥 2 𝑑𝑥
2)
(𝑥𝑒 𝑥 +1)𝑑𝑥
0
3)
5)
𝑒𝑥
2𝜋
4
cos 𝑥 𝑑𝑥
5
7xtan
−1
4𝑥 3 sin 2𝑥 𝑑𝑥
4)
𝜋
2
1
2
3𝑥 𝑑𝑥
6)
𝑠𝑖𝑛−1
0
31
2
𝑥 𝑑𝑥
3
Integrate the following integrals using the partial fractions method of integration. A
calculator is not required to solve these problems.
4
7)
0
9)
1
𝑑𝑥
𝑥2 + 𝑥 − 6
15𝑥
𝑑𝑥
8𝑥 3 − 24𝑥 2 − 72𝑥 − 40
8)
Note: One zero of the function
𝑓 𝑥 = 8𝑥 3 − 24𝑥 2 − 72𝑥 − 40
is 5
8
𝑑𝑥
𝑥 2 − 2𝑥 + 1
10)
Note: Account for the repeating
part of this function.
8
11)
1
4𝑥 + 1
𝑑𝑥
90𝑥 2 − 45𝑥
32
5𝑥 2 + 3𝑥 + 7
𝑑𝑥
3𝑥 2 − 𝑥 − 10
Integrate the following integrals using the improper integrals. A calculator is
not required to solve these problems. You may have to use L’Hopital to solve
these problems. Watch out for undefined values!
∞
12)
0
6
1
𝑑𝑥
𝑥2 + 4
13)
−∞
1
𝑑𝑥
𝑥3
∞
7𝑒 −𝑥 𝑑𝑥
14)
0
16)
10
∞
17)
5𝑥𝑒 −𝑥 𝑑𝑥
0
−∞
33
2
𝑑𝑥
𝑥−8
AP Multiple Choice Exercises
1) What is the sum of the following integral?
A)
B)
3𝑥
𝑑𝑥
10𝑥 2 − 17𝑥 − 20
−30𝑥 2 − 60
(10𝑥 2 − 17𝑥 − 20)2
𝑙𝑛 5𝑥 + 4
12
66
15
165
(2𝑥 − 5)
+c
12
C)
𝑙𝑛
(5𝑥+4)165
15
(2𝑥−5)66
D) 𝑙𝑛 5𝑥 + 4
+c
12
165
15
66
(2𝑥 − 5)
+c
E) The function is not integrable.
http://www.panempropaganda.com/news/2012/5/1/katniss-and-peetas-festival-of-victory-the-most-elaborate-vi.html
34
2) Katniss is frolicking like a kitten along the x-axis, looking for catnip at a rate, in
gigameters per nanosecond, modeled by the function 𝑓 𝑡 = 𝑡𝑐𝑜𝑠𝑡, 0 ≤ t ≤ 𝜋. Upon
reaching the catnip, Peeta steals it, because he mistakes it for pita bread and runs in
the opposite direction. How far from her starting point has she frolicked when Peeta
takes the catnip and she begins frolicking backwards to get it? You may not use a
calculator for this problem.
𝜋
2
B) -2
C)
𝜋−2
2
D)
𝜋+2
2
35
http://www.loupiote.com/photos/6567820901.shtml
http://www.myhungergames.com/the-hunger-games-console-games
E) 3
http://www.catclaws.com/Certified-Organic-Catnip-8-oz-Bag/productinfo/1480/
A)
3) Which of the following integrals are determinate?
I.
2
1
𝑑𝑥
−5 8𝑥−24
II.
∞ 1
𝑑𝑥
15 𝑥 2 +60
III.
10 𝑙𝑛𝑥
𝑑𝑥
−1 𝑥
A) I and II only
B) III only
C) I and III only
D) I, II, and III
E) II only
http://nyulocal.com/on-campus/2012/03/23/girl-on-fire-hunger-games-review/
36
4) Solve the following integral.
𝑥2
5𝑥𝑎𝑟𝑐𝑠𝑖𝑛 12dx
A)
5𝑥 2
𝑥2
arcsin
+
2
12
144 − 𝑥 4
+𝑐
2
B)
5𝑥 2
𝑥2
arcsin
−
2
12
𝑥 4 − 144
+𝑐
2
C)
5𝑥 2
𝑥2
arcsin
−
2
12
144 − 𝑥 4
+𝑐
2
30
D)
E)
𝑥4
+𝑐
1 − 144
http://www.fashionresister.com/2012/11/occ-metallurgy-super-nswf.html
𝑥2
−5𝑥𝑎𝑟𝑐𝑐𝑜𝑠
+𝑐
12
37
4
9𝑥𝑙𝑛 𝑥 𝑑𝑥
Evaluate the following integral.
1
A) 8ln4-3.75
B) -6.75
C) 72ln4+33.75
D) 72ln 4-33.75
E) 72ln4-3
http://www.dragoart.com/tuts/10222/1/1/how-to-draw-hunger-games,-the-hunger-games-logo.htm
38
AP Free Response
During Katniss’s journey through the Integral Games, several other players try to kill her
with their superior calculus. In order to escape her threatening competitors, Katniss
decides to disturb a tracker jacker hive. Although she succeeds in repelling her attackers,
she is stung multiple times. Luckily, four hours after she is stung, Rue removes the
stingers and applies anti-venom. The rate at which the tracker jacker venom is entering
𝒕𝒆𝒕
𝟑𝒆𝟒
Katniss’s body, in milliliters per hour, is modeled by the function 𝒇′ 𝒙 = 𝟐 − 𝟖 + 𝒕,
0 ≤ t ≤ 4. The rate at which the anti-venom neutralizes the venom, in milliliters per
𝟑
𝟏𝟗𝟎
hours, is modeled by the function 𝒈′ 𝒙 = 𝟖 𝒕𝟐 − 𝟗𝟔 𝒕, t ≥ 4.
𝟒 𝒕𝒆𝒕
𝟏
integral 𝟒−𝟎 𝟎 ( 𝟐
𝟑𝒆𝟒
𝟖
(a) Evaluate the
−
+ 𝒕)𝒅𝒙 . Using correct units explain the
meaning of your answer.
(b) At what time after Katniss gets stung has all of the venom in her body neutralized?
(c) If Rue did not show up to save Katniss, at what time would Katniss have died? Note:
20 ml of venom is deadly.
http://thehungergames.wikia.com/wiki/Tracker_jacker
39
Works Cited
"Archimedes." Ancient Greece. N.p., 2003-2012. Web. 03 June 2013.
"Archimedes of Syracuse." JOC/EFR, Jan. 1999. Web. 03 June 2013.
Bourne, Murray. "Applications of Integration." Interactive
Mathematics. N.p., 24 Aug. 2012. Web. 03 June 2013.
Burt, Brandon. "Re: What Are the Uses of Improper Integrals/calculus?" Web
log comment. Answers.yahoo.com. Yahoo Answers, 2008. Web. 3
June 2013.
Divo, Eduardo. Integration by Parts Applications in Engineering. Rep. N.p.:
n.p., 2009. Print.
Khamsi, Mohamed A. "Convergence and Divergence of Improper Integrals."
Convergence and Divergence of Improper Integrals. SOS
Mathematics, 3 Dec. 1996. Web. 03 June 2013.
Petrov Petrov, Yordan. "The Origins of the Differential and Integral Calculus 1." Math10.com. N.p., 27 Sept. 2005. Web. 03 June 2013.
Simmons, G. "History of Calculus." N.p., 1985. Web. 03 June 2013.
40