L*Hopital, Integration by Parts, and Partial Fractions

Report
L'Hôpital, Integration
by Parts, and Partial
Fractions
Jeanne Tong & Marisa Borusiewicz
Table of Contents
Slide #
Rapunzel’s Real World Applications
Mickey Mouse’s Clubhouse of Fame
Mulan’s teachings of Integration by Parts
Integration of Partial Fractions with Nemo
L'Hôpital’s Rule
Story Time!
Cinderella’s Analytical Example
AP Multiple Choice with Pinocchio
Simba’s Solution
AP Conceptual Problem
AP Conceptual Problem Solution
AP Level Free Response with Solution
Graphical Problem
3
4
5
6
7
8
9
10
11
12
13
14
15
Coloring Page
Works Cited
16
17
*All pictures used are copyrighted by Disney.
The Wonderful World
of Real Life Application
You may be wondering if integration
by parts, partial fractions, or L’Hopital’s rule
have any practical uses. Of course, the
answer is yes! Integration by parts and
partial fractions are variations of standard
integration which has a multitude of
purposes. One of the major purposes is
integrating the rate of growth of Rapunzel’s
hair. She can never seem to remember how
long her hair is, but by using an equation for
the rate of growth she can easily just
integrate using by parts to solve for the
length.
Other minor uses can be found in the
fields of physics, engineering, architecture,
business, and chemistry. Modeling the change
in mass, energy or momentum on both a
micro- and macroscopic scale with equations
allows a physicist to be able to study the
interaction between objects in the universe.
Integration is also useful when it comes to
the motion of waves. Vibration, distortion
under weight and fluid flow, such as heat
flow, air flow, and water flow all involve
integration. These may be helpful to
engineers designing planes, ships, pipe
systems, submarines, or magic carpets.
Architects might need to consider these ideas
when designing buildings, bridges, or
structures with unequal forces acting upon
it. Chemists utilize integration when finding
the pH of titrations. During these
experiments pH is often plotted and a curve
is fitted to the data. Integrating these curves
can be used to predict and analysis the pH
trends. When modeling regression curves,
analyzing population, or studying the
kinematics of the cell process, integration can
but used to help produce a model or curve
and L’Hopital can be used to find the bounds
and limits.
Johann & Jacob Bernoulli
Guillaume de L'Hôpital
Born into a noble French family in Paris,
L'Hôpital is associated with the L'Hôpital
rule. After abandoning his military
career, L'Hôpital continued to pursue his
interests in the mathematical field. In
1691, L'Hôpital met Johann Bernoulli.
Johann became L'Hôpital’s instructor,
giving him private lectures. Later, in
1964, L'Hôpital made a deal with
Bernoulli for an annual payment of 300
Francs in exchange for Bernoulli’s latest
mathematical discoveries (essentially it
was a bribe). Eventually, after the
creation of L'Hôpital’s rule, Bernoulli
was credited because he was unhappy
with the unjust publicity of L'Hôpital’s
work. The L'Hôpital rule is the epitome
of limits in indeterminate forms. When
a limit is indeterminate always
remember this key phrase: “take it to
the Hospital!”
Use this formula to
solve for integrals
that resemble the
method of
integration by
parts:
udv

uv

vdu


HINTS AND TIPS: to easily
solve integrals using by
parts, let u equal an easily
differentiable function and
let dv equal a function that
can be easily integrated!
If you see an integral with a
denominator that looks
relatively easy to factor, this
means use the partial fractions
method!
Here are a few examples
of when you should use
the method of integration
for partial fractions:
x
x
1
2
 5x  6
dx
(5 x  3)
2
 2x  3
dx

x
1
2
 2x 1
4x 1
x  x2
2
( 4 x  5)
x
2
x2
dx
dx
dx
x
2x  3
2
 3x  2
dx
lim
x c
f(x)
g(x)
 lim
x c
f'(x)
g'(x)
Indeterminate forms when
taking the limit:
0  (  )

(  )
, ,
,
,
0 

(  ) (  )
If you encounter one
of these indeterminate
forms, this is a huge
clue directing you to
use L'Hôpital’s rule!
An indeterminate
form tells us that no
specific limit is
guaranteed to exist or
the limit cannot be
found.
In order to use
L'Hôpital’s rule, f
and g must
differentiable
functions
If it’s
indeterminate,
take it to the
Hospital!
STORY TIME!
Once upon a time there was a lovely
princess. But she had an enchantment upon
her of a fearful sort which could only be
broken by love's first kiss. She was locked
away in a castle guarded by a terrible firebreathing dragon. Many brave knights had
attempted to free her from this dreadful
prison, but non prevailed. She waited in the
dragon's keep in the highest room of the
tallest tower for her true love and true love's
first kiss.
Analytical Example (with solution)
Find the solution to the indefinite integral:
Although this integral may look a little intimidating at first, partial fractions can make it a rather
simple integral. Partial fractions can be useful when the denominator is easily factorable.
(Here the denominator has been factored )
From this factored form , we can divide the fraction into the sum of two
fractions, assigning the numerators separate variables, in this case A and B.
Allowing this new from to be equal to the original, multiply both
the new and the original by the denominator of the original.
Distributing and simplifying , we find that the we are left with:
In order to solve for the variables A and B, the terms that they are contained within, must
simplify to zero. Substitute in the values that will make the terms equal to zero and solve for
both A and B.
B 
4
3
We can substitute the values for A and B into the factored form of the integral. This allows us to
divide the integral into 2 separate integrals
From this form we can in integrate as normal and find the indefinite integral
Don’t forget to use usub with this one!
 8  1 
 4
    ln( 2 x  1)     ln( x  2 )  C
 3  2 
 3
4  2x 1
ln 
C
3  x2 
Don’t forget +C!
AP Level Multiple Choice
Find
A)
x cos x  2 x sin x  2 cos x  C
B)
 x cos x  2 x sin x  2 cos x  C
C)
x sin x  2 x sin x  2 cos x  C
D)
x
2
2
2
3
 cos x  C
3
E)  x 2 cos
 udv
x  2 sin x  2 x cos x  C
 uv   vdu
Correct answer: B; integration by parts must be done twice
A: positive and negative signs are incorrect
Abracadabra! Reveal the answers!
C: student may have mistakenly multiplied (u)(dv) instead of uv
D: completely wrong answer; student didn’t use the by parts
formula
E: multiplied by the wrong variables when the student did
integration by parts the second time, which led to incorrect
integration using the by parts formula
Simba’s Solution
DON’T FORGET!
 udv
 uv   vdu
Remember
that u and v
must be
functions of
x and be
continuous
derivatives
Find
 x cos x 
2
 x cos x 
2
 2 x cos
u  2x
du  2
xdx
v  sin x
dv  cos x
 x cos x  2 x sin x  2 sin xdx

2
Choice B
 x cos x  2 x sin x  2 cos x  C
2
HINT: You must
do integration
by parts a
second time in
this problem!
AP Conceptual Problem
0
3
6
9
f(x)
1
6
3
7
f’(x)
4
2
-5
-12
g(x)
-2
-1
8
2
g’(x)
7
-3
1
10
Using the table above:
6
Solve

0
6
f ' ( x ) g ( x ) dx
given

f ( x ) g ' ( x ) dx  20
0
Take a moment to solve this AP conceptual
problem and enjoy the soundtrack!
AP Conceptual Problem
Solution
6
6
Solve

f ' ( x ) g ( x ) dx given

f ( x ) g ' ( x ) dx  20
0
0
Remember:
 udv
 uv   vdu
Using the by parts formula above, set the proper functions equal
to u and dv. In order to get du, take the derivative of u. To get v,
you must take the integral of dv.
u  g ( x)
v  f (x)
du  g ' ( x )
dv  f ' ( x )
6
g ( x) f ( x) 

f ( x ) g ' ( x ) dx
0
6
You are given that 
f ( x ) g ' ( x ) dx  20
0
 g ( x ) f ( x ) 60  20
Follow normal integration rules for definite integrals doing:
b

f ' ( x ) dx   f ( x ) a  f ( b )  f ( a )
b
a
Now just plug in the bounds and look on the table for values to solve!
[ g ( 6 ) f ( 6 )  g ( 0 ) f ( 0 )]  20
( 8 )( 3 )  (  2 )( 1)   20
 6
.
Aladdin took the princess from the palace through the city The speed
of Aladdin’s magic carpet is modeled by the function
If the trip took one hour and a half what was his average speed?
(measured in miles per minutes)
b
1

ba
For this problem we must start with the basic integral
for an average value.
f ( x ) dx
a
90
1
 ( 5000
90  0
5000
4
x e
2 x
) dx
Plug in for the bounds and the equation into the
integral
0
90
 (x
90  0
4
e
2 x
By the rules of integration we can remove the 5000
from within the integral and multiple it after we
integrate.
) dx
0
 udv
 uv 
 vdu
From here we must use by parts to integrate. You
may use the traditional UV from of integration, but
for this example we will complete this with table
method.
In order to use table method, it is most effective if
you pick a u which is easily differentiable and a dv
which you can easily integrate. From here, list the
derivatives of u and the anti-derivatives of dv. Every
other derivative of u must be negative. Integrating
the original function can be done by combining the
derivatives and anti-derivatives in a diagonal fashion
as shown to the left. Each diagonal represents a term
that will be added together to form in the integral.
5000
90  0
90
 (x
4
e
2 x
) dx
Finally, we can evaluate from 0 to 90 minutes and
find the average speed!
0
This is really fast for a magic carpet! At this rate
he can literally show Jasmine the whole world
in just about 25 days!
mi/min
When the problem asks for average value over a time
interval, you must have b 1 a in front of the integral
Graphical Problem
lim
x 0
sin x

x
sin( 0 )

0
0
0
Use L'Hôpital’s rule because when you take the limit as x
goes to 0, you get an indeterminate form of 0/0. Take the
derivative of the top and bottom separately…now the limit
as x approaches 0 is 1.
lim
x 0
This is a graph of
cos x

cos( 0 )
1
1

1
1
You can see that as the
limit goes to 0, the graph
approaches 1
sin x
x
y  x
y ' (0)  1
y  sin( x )
y ' (0)  1
As you can see both of these graphs pass through zero, but
the derivatives (slopes) equal 1, which allows you to find the
limit of the function sin x as x goes to 0.
x
One of the bugs Timon pulls out of the log during
'Hakuna Matata' is wearing Mickey ears.
Time for a coloring
break!
Use a lot of blue! Did you know the color blue
has a calming effect because it stimulates the
release of hormones that low blood pressure?
In the Disney movie Hercules is the son of the two
gods, Zeus and Hera, but, according to traditional
Greek mythology, he is the son of Zeus and the mortal
woman Alcmena .
While this page may be skipped, it is advised not to. I know the calculus is very exciting,
but it has been proven that taking a break from learning increases the amount of
information you retain by forcing you to refocus your thoughts. So color on my friends!
Works Cited
•
•
•
•
•
•
All pictures are copyrighted by Disney.
http://www.physicsforums.com (accessed: 5/12/12)
http://www.intmath.com/applicationsintegration/applications-integrals-intro.php (accessed:
5/15/12)
thickclouds.com (accessed: 5/17/12)
http://wwwmath.ucdenver.edu/~wcherowi/courses/m4010/s08/csbernoul
li.pdf (accessed: 5/13/12)
Rob Larson: Analytical Calculus 8th edition textbook
Analytical Exercises
1. Integrate
2. Integrate
3. Integrate
 x sec
2
xdx

x e dx

x ln xdx
2
2x
2
 /2
4. Integrate
 e cos xdx
x
0

5. Integrate
 3 xe dx
3x
0
Analytical Examples
6. Integrate
7. Integrate
1
x
 6x  8
2
x
6x
2
x2
dx
dx
x 2
2
8. Integrate
9. Integrate
10. Integrate
x
x
2
 7 x  30
dx
6 x  20
3
 4x  4x
2
dx
2 cos x sin x
 cos
2
x  cos x
dx
Analytical Examples
2x
 cos tdt
11. Integrate
0
lim
x 0
2x
x 9
2
12. Integrate lim
x 3
13. Integrate
14. Integrate
x  6x  9
3
lim x
1/ 2 x
x 
lim
x 0
15. Integrate lim
x 0
xe
4x
x
1  cos( 2 x )
sin 2 x  sin x
x
3
AP Practice Multiple Choice
Solve the integral:
1.

x 4
2
2
2
x
x e dx
2. lim
x 2
0
x
x5
2
a.
 2e
2
b.  2 e  2
2
c.
 4e  2
2
d. 2 e  2
8 2
e
e.
3
2
a. 1
b.
2
3
c. 0
4
d.
e.
5
1
3
3. Given that
f ( x)  x  2
a. 2
b.
c.
2
e
2
e 1
d. 0
e.
underfined
2
, find
lim
x 0
f ( x)  2
e 1
x
4.
x
2
dx
2
5.
4
x4
x
1
a. 1 ln x  2  C
x2
2
b. 1 ln x  2  C
2
x2
c.
d.
1
ln
x2
4
x2
1
x2
ln
4
e. 
1
4
x2
ln
a.
b.  3 ln 3  7 ln 2
c.
ln
3
x2
x2
C
2
3
3
C
x
3 ln 3  5 ln 2
7
C
2
2
d.  7 ln 3
e.  3 ln 2  7 ln 3
AP Practice Free
Response
Let f ' ( x )  x 2  3 x 2 ln x , for which f(x)
is continuous and differentiable
with an initial condition that f(0)=0.
a. Use antidifferentiation to find f(x)
b. Find lim
x 1
f ( x)
x 1
2
c. Determine if any maximums or
minimums exist on f(x) on the
interval 0<x<2

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