CHAPTER FIVE: THE INDEFINITE INTEGRAL AND BASIC DIFFERENTIAL EQUATIONS

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CHAPTER FIVETHE INDEFINITE INTEGRAL AND BASIC
DIFFERENTIAL EQUATIONS
Hare Krsna Hare Krsna Krsna Krsna Hare Hare Hare
Rama Hare Rama Rama Rama Hare Hare Jaya Sri Sri
Radha Vijnanasevara (Lord Krsna, the King of Math
and Science) KRSNA CALCULUS PRESENTS

• February 2, 2002
• Released by Krsna Dhenu

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HARE KRSNA! SVAGATAM!

• This chapter about the indefinite integral. In
  chapter 4, we dealt with definite integrals, in
  which we had limits to deal with. However, with
  indefinite integrals, we dont have limits, i.e.
  we dont have the a or b in the integral sign.
• You are expected to know ALL the rules of
  differentiation you learned from chapter 4.
• You will also deal with the infamous words
  differential equations. Differential equations
  (commonly referred as diff-eqs) basically an
  equation that has a derivative in it. More on
  that, when we get there.
• For college calculus I students, this is the
  final chapter for you. You might go into a little
  bit of chapter 6, but that is very questionable.
  Start from the midpoint of this chapter, since
  you are not responsible for differential
  equations.

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DIFFERENTIAL EQUATIONS

• A differential equation is an equation that
  contains a derivative. For example, this is a
  differential equation.
• From antidifferentiating skills from last
  chapter, we can solve this equation for y.

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THE CONCEPT OF THE DIFFERENTIAL EQUATION

• The dy/dx f(x) means that f(x) is a rate. To
  solve a differential equation means to solve for
  the general solution. By integrating. It is more
  involved than just integrating. Lets look at an
  example

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EXAMPLE 1

• GIVEN
• Multiply both sides by dx to isolate dy. Bring
  the dx with the x and dy with the y.
• Since you have the variable of integration
  attached, you are able to integrate both sides.
  Note integral sign without limits means to
  merely find the antiderivative of that function
• Notice on the right, there is a C. Constant of
  integration.

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C?? What is that?

• Remember from chapter 2? The derivative of a
  constant is 0. But when you integrate, you have
  to take into account that there is a possible
  constant involved.
• Theoretically, a differential equation has
  infinite solutions.
• To solve for C, you will receive an initial value
  problem which will give y(0) value. Then you can
  plug 0 in for x and the y(0) in for y.
• Continuing the previous problem, lets say that
  y(0)2.

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Solving for c.
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SLOPE FIELDSAP CALCULUS MATERIAL ONLY

• We just solved for the differential equation
  analytically (algebraically). The slope field
  (also known as vector field and directional
  field) will give us a qualitative analysis.
• The graph shows all the possible slopes in the
  form of a field.
• The arrows show the basic trend of how the slope
  changes. Using the initial condition, you can
  draw your solution.
• For the previous example, the slope field will be
  very simple to draw.

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SLOPE FIELD FOR EXAMPLE 1

• Notice how slope field TRACES the tangent lines
  of points from the antiderivative from various
  constants.
• For the curve that is relevant with the correct,
  in our last problem C2, connect those particular
  tangent lines and heavily bold it.
• I drew this by hand, so please forgive my
  sloppiness with the slopes. _/\_ ?

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SLOPE FIELDS

• The previous was so easy that a slope field was
  really not required.
• However, there are many differential equations
  that will not yield easily to form such a slope
  field.

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HOW TO DRAW SLOPE FIELDS

• Consider dy/dx-2xy. This is the formula for
  SLOPE
• To find the slope, you need both an (x,y)
  coordinate. For example, if you use (1,-1), then
  the slope (-2)(1)(-1)2.

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DRAWING SLOPE FIELDS

• Start from (1,-1) and make a small line with the
  slope of 2. (Remember in high school, when you
  did lines, how did you do slope? Difference in y
  over difference in x).
• Thus, the solution of the differential equation
  with the initial condition y(1)-1 will look
  similar to this line segment as long as we stay
  close to x-1.

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DRAWING SLOPE FIELDS

• However, simply drawing one line will not help us
  at all. You have to draw several lines. This what
  gets the Durvasa Muni out of the calculus
  students!
• Then connect the lines horizontally to fit a
  curve amongst the tangent lines. These lines are
  formed from various C values.

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SLOPE FIELDS

• This topic of slope fields will be discussed
  highly in a college differential equations
  course. The AB Calculus exam, since 2002, has
  included slope fields in the curriculum, they
  have to know just as much about slope fields as
  BC Calculus.
• The college calculus teachers generally like to
  skip over such topics of differential equations,
  even the easy ones like the first example.
• Lets consider the last example dy/dx-2xy. Say
  we were the 2001 graduating class (thats my
  graduating class ?) and we didnt learn slope
  fields. How would we such such an equation since
  there is a y there.

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SEPERATION OF VARIABLES

• Such equations are known as separable
  differential equations. The way to go about
  solving such equations (raksasas lol ?) is to
  round up your y terms with dy and round up your x
  terms with dx.
• When integrating dy/y, remember the derivative
  of ln y is 1/y. Therefore the integral of 1/y is
  ln y.

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INTIAL VALUE PROBLEM

• Lets say that y(1)-1. We can find C that way.
• And finally, your exact answer.

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AUTONOMOUS SEPARABLE DIFFERENTIAL EQUATIONS
(A.S.D.E.)

• A differential equation that is autonomous means
  that the derivative does not depend on the
  independent variable. For example, The equation
  below is an autonomous equation. Notice that
  there is no x involved.

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SOLVING A.S.D.E

• You can still separate the y and bring dx to the
  right.
• The process is the same. ?

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POPULATION GROWTH

• The rate of the population for New Vrndavana is
  shown by the differential equation, dy/dtky. If
  k is the population constant, let k1. If t is
  measure in years and if y(0)200, then what is
  the predicted population in 2 years?

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POPULATION GROWTH

• Since we were given rate, we have to find the
  population by solving the differential equation.
• You can use separation of variables to solve this
  autonomous equation. Note how I solve for y.

• GIVEN
• Separate variables
• Integrate
• Exponentiate to undo the natural log on the left.
• ekt and eC can be separated as such due to the
  laws of exponents.
• eC itself is a constant so you can rename that
  number as C.

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SOLVING FOR C.

• Not too difficult, since the problem said
  earlier, that k1, we dont need to keep the k in
  there. Since, t0, the exponent iteself will be
  zero, therefore e01, thus y(0)C or C200.

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PREDICTING THE POPULATION

• You can predict the population, by simply
  plugging t2 in y(t) and get the answer.
• 1477 people is the answer.
• See how Lord Caitanya was true when He says,
  Every town and village! New Vrndavana
  population increases! Jaya! ?

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CONCLUSION TO DIFFERENTIAL EQUATION CONCEPTS

• After seeing the population growth problem, it is
  best stop here for the differential equations
  portion of this chapter. This topic is covered in
  the AB and BC Calculus exams.
• Knowing that the curriculum changes a lot each
  year, when necessary, I will change the
  presentations to fit their standards.

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INDEFINITE INTEGRALS

• CALCULUS 1 STUDENTS, YOU START THIS SECTION
• An indefinite integral came up frequently in the
  differential equations part of the chapter. It is
  the integral with no limits. It is used to merely
  antidifferentiate functions.

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INTEGRALS THAT SHOULD BE KNOWN
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U-SUBSTITUTION

• If you read the conclusion of Chapter 5, where I
  threw my opinions, you will notice how I
  described the reason why integral calculus is
  very long. The only reason why is because the
  integral of a product or quotient of two
  functions is not equal to the product or quotient
  of the integrals of the functions, respectively.
• Therefore, a good amount integral calculus is
  antidifferentiation. There are many ways to
  integrate functions. If you are a calculus I or
  AB student, then this will be the only way you
  will learn this semester. If you are a BC
  student, you will learn many ways, this being
  your first way.
• Again, memorize that table on the last slide with
  all the integrals!!!!!!

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INTEGRATION BY SUBSTITUTION (u substitution)

• Consider this function. You can use the binomial
  theorem, expand it and integrate each term piece
  by piece. Very tedious but doable.

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U-SUBSTITUTION

• However, there is no way you can break something
  like this down and integrate easily. Therefore,
  with the rules that we learned so far, you cannot
  integrate such a function.

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SUBSTITUTING

• This function could be a bit easier as well as
  pleasing to look at (but Krsna is most pleasing
  to look at), if you substituted a variable and
  integrated that way. Like the chain rule, lets
  use the argument to be the u.

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CHOOSING U

• GIVEN INTEGRAL
• Defined u(x).
• Substituted u into the integral.
• So we can integrate, right?? WRONG! Look at the
  variable of integration.. Its dx. We didnt take
  into account this. Remember the Reimann sum? It
  was the function times the really small change in
  x, namely dx. Therefore, we have to find a dx
  substitute.

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A dx substitute

• To get dx, differentiate u, thus, you will get
  du/dx. Multiply both sides by dx.
• Note, that we have 2dx du. We dont want a 2dx,
  we just want a dx. So we simply divide both sides
  by 2.
• Then plug du/2 in for dx in the integral. Now
  integrate! Note, how I was able to pull the ½ out
  of the integral.

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LETS TRY SOMETHING MORE CHALLENGING

• GIVEN
• Name u
• Notice that we solved for xdx. We gave a name for
  x2-3, but needed a name for xdx, therefore, we
  only solved for what we needed. This is a key
  idea in doing these u-substitution problems.
  Solve ONLY for what you need.

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U-SUBSTITUTION

• Lets try another one. Matter of fact, lets
  invent the integral of tan x!
• You have to set it as sin x/cos x.
• Let u cos x. thus, -du sin x.
• DO NOT LET u sin x in this example, because du
  cos x dx, and we dont have it. In fact, we
  have dx/cos x if you look at it.
• Remember rules of logs
• -lna ln1/a

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EVALUATING DEFINITE INTEGRALS USING u-substitution

• In terms of integrating the function, itself, the
  rules are the same, however, there are two ways
  to evaluate it at the limits.
• GIVEN

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DEFINITE INTEGRAL

• Lets worry about the function first. The
  expression under the square root looks like a
  good place to call that u. Another thing, I
  should have emphasized is that look for
  derivative similarities. You see x3 with a
  constant. You know that if you differentiate
  that, youll get just an x2. ALWAYS look for
  derivative similarity!

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EVALUATING THE LIMITS

• If you want to use u, then evaluate u at both
  limits. In this case u(0)0 and u(2)4.
• You could also, after integrating the function
  with respect to u, you can replace the u(x) back
  into and then evaluate using the original limits

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Why is there no C in the definite integral?

• If you look at the fundemental theorem of
  calculus, F(b)-F(a), you will see that both C
  values cancel. It becomes immaterial whatever the
  function has a constant added to it or not.
• Whenever you not definite (indefinite), then add
  Lord Caitanya (C) to everything. Then youll
  have infinite answers ?!!! Jaya Sri Sri Gaura
  Nitai!
• Touch Lord Caitanya Mahaprabhus Feet and just
  wait for a moment!!!!! Seriously, take your mouse
  and touch His feet!

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SUMMARY for Diff-Eqs.

• A differential equation is an equation that
  contains a derivative in it. There are infinite
  solutions, but with the help of an initial value
  you problem, you can solve for an arbitrary
  constant, to help you get the final equation.
• You can solve it analytically using integration,
  or qualitatively by slope fields.
• Slope fields (also known as vector or direction
  fields) show the general trend of solutions of
  the differential equation with various C values.
• An autonomous equation is a differential equation
  which does not depend on the independent
  variable. For example, dy/dxf(y) is autonomous.
• Exponential growth can be identified with the
  equation f(t)Cekt.

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INTEGRATION

• An indefinite integral is an integral without
  limits. They are the antiderivative plus an
  arbitrary constant.
• There is a very important table of integrals you
  SHOULD memorize!! In addition, memorizing
  derivatives are very very important! If you are
  not proficient in integration, DO NOT GO PAST
  THIS CHAPTER AT ALL. Chapters 7 11, deal with a
  great deal of integration. Chapters 12-YTBK deals
  with function with many variables and will
  redefine integration in a very different
  perspective.
• A helpful technique is u-substitution. Use this
  technique if you are able to see the a function
  f(x), with its derivative in a different form,
  multiplied together, then you should use this. Be
  sure to put the u in the right place.

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A STEP BACK

• Well, I admit, just for only five chapters, we
  talked about a lot of calculus! We started from
  the basic definition of limit, and then we got
  the derivative with its applications and
  concepts. Then, we got to area under the curve.
  That Reimann sum got us the idea of integration.
  From their, the link of derivative and integral
  was made.
• Remember when I told you in Chapter 1 conclusion
  that calculus is all algebra, geometry, and
  trigonometry learned in high school, with three
  new things 1) limits 2) infinity and 3) 0/0. We
  know what slope is when we studied lines in high
  school. We merely took the limit as the change in
  x got really really really small. That became
  known as the derivative. We saw when taking the
  limit as the amount of rectangles went to
  infinity, or as the width of the rectangles got
  close to 0, then we developed the integral. We
  will use the idea of limits more throughout this
  course.

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NEED HELP?

• Call me 716-645-4416. Ask for Krsna Dhenu!
• E-mail me vedicger108_at_hotmail.com
• Till then
• Jaya Sri Krsna Caitanya Prabhu Nityananda
• Sri Advaita Gadadhara Srivasadi Gaura Bhakta
  Vrnda
• Hare Krsna Hare Krsna Krsna Krsna Hare Hare
• Hare Rama Hare Rama Rama Rama Hare Hare

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CREDITS

• SOS Math webpage was used for the slope field
  drawing techniques.
• Mr. J. Trapani
• Mr. G. Chomiak
• Dr. W. Menasco
• Dr. N. G. Goodman
• ISKCON B.B.T. for Lord Caitanya Mahaprabhus
  picture.
• The soundfile came from yours truly ? (Krsna
  Dhenu) with a Ratha Yatra Bengali style kirtana.
  Visit my music page for more.

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END OF CHAPTER FIVE