CHAPTER EIGHT: INTEGRATION TECHNIQUES: - PowerPoint PPT Presentation

About This Presentation
Title:

CHAPTER EIGHT: INTEGRATION TECHNIQUES:

Description:

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) – PowerPoint PPT presentation

Number of Views:163
Avg rating:3.0/5.0
Slides: 55
Provided by: Nish151
Category:

less

Transcript and Presenter's Notes

Title: CHAPTER EIGHT: INTEGRATION TECHNIQUES:


1
CHAPTER EIGHT INTEGRATION TECHNIQUESAS IT IS
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
  • Released by Krsna Dhenu
  • September 28, 2002
  • Edited October 7, 2003

2
WELCOME BACK!!!
  • Hare Krsna everyone!
  • Please take a moment to sigh before moving onto
    the next slide.
  • This chapter is rated one of the most difficult
    chapters. This contains a great amount of
    material.
  • It is strongly suggested that you do NOT start
    this chapter without the strong backgrounds of
    derivative, integral, function behavior, and
    algebra.

3
WHAT IS THIS CHAPTER ABOUT?
  • Notice when we did derivatives, we only spent one
    whole chapter on how to compute the derivative
    (Chapter 2)
  • We spent time using Chapter 3 to do real-world
    examples using derivatives.
  • We know how to differentiate ANY function.
  • Integrals, however, are more difficult, since
    there are many rules involved. There is no one
    method of computing integrals.
  • This chapter is completely devoted to different
    methods of integration. It is important that you
    take this chapter nice and slowly so you can
    build it in.
  • Also, you must develop critical thinking in this
    chapter. Without it, this chapter will become
    impossible.

4
INTEGRATION WHAT DO WE KNOW SO FAR?
  • In terms of getting integrals, we know how to do
    basic anti-differentiation with power functions,
    trig functions and exponential functions.
  • We started some techniques using u-substitution
    to solve integrals. More or less, a reverse
    chain rule.
  • We also did area, volume, and other physical
    applications using the definite integral.
  • This chapter is devoted to the indefinite
    integral.

5
1) U-SUBSTITUTIONREVISITED
  • Just to freshen your minds, lets do a
    u-substitution problem. There is a little catch
    to it
  • For this problem, it is preferable to pick u
    x2, since du can never be in the denominator.
  • Use algebra to give a name for x1. Since xu-2,
    then x1 would be u -21 or u-1.
  • After putting everything in, do the integration.
  • Note that in the final answer, the constant 2
    does not need to be there, since C is more
    general.

6
PRODUCT RULE FLASHBACK
  • Remember back in Chapter 2, we mentioned the
    product rule? Looked like the following

In regular form. Or differential form.
7
2) INTEGRATION BY PARTS
  • If you play around with the differentials, you
    will get the following.
  • If you integrate both sides, you get what is
    known as the INTEGRATION BY PARTS FORMULA

8
EXAMPLE1
  • Integrate xexdx.
  • STEP 1
  • Pick your u and dv.
  • TRICK L.I.P.E.T. This determines what u should
    be initially. Logarithm, Inverse, Power,
    Exponential, and Trig functions.
  • In this problem, a power function and an
    exponential function are present. Since power (P)
    comes before exponential (E), u should equal x.
  • From the u, find du. From the dv, find v.
  • Therefore ux, then dudx! Dont forget that.
  • Simply plug this into the parts formula.

9
EXAMPLE1
  • I integral to be solved.
  • Plug in u, v, du, and dv.
  • If the integral on the right looks easy to
    compute, then simply integrate it.
  • Dont forget the C!

10
EXAMPLE2
  • Integrate the function

11
EXAMPLE2
  • Step 1 Find the u and dv
  • An exponential and a trig function is present.
  • E (exponential) comes before T (trig).
  • Use the exponential function for u.

12
EXAMPLE2
  • Step2 Plug u, du, v and dv into the formula
  • Simplify
  • If the last integral is easy to compute, compute
    it.
  • However, it is not easy. In fact, we are in more
    mess than we started in. Looks like we have to
    use the integration by parts rule again.

13
EXAMPLE2
  • Use integration by parts again. Since there is an
    exponential function, call that u. After
    finding u, du, v, and dv, plug those values in
    the Parts equation and see what happens
  • We have even more of a messbut wait! The
    integral of exsin(x) is supposed to be equal to I
    (the integral we wanted to solve for in the first
    place!!).
  • So we can replace the integral of exsin(x) with I
    and add it to both sides.
  • You will see that it works out.
  • Always add the constant ?

14
RULE OF THUMB WITH INTEGRATION BY PARTS PROBLEMS
  • Always pick the right u. If the problem is
    getting really difficult, maybe you picked the
    wrong u. Just like in u-substitution. You had
    to pick the right u to work with the problem.
  • If the integral on the right does not look easy
    to compute, then do integration by parts for that
    integral only.
  • If you see the resulting integral looks like the
    integral you are asked to solve for in the first
    place, then simply combine the two like integrals
    and use algebra to solve for the integral.

15
3) TRIGONOMETRIC INTEGRALS
  • You will always get the situation of funny
    combination of integrals. Trig functions are as
    such that you can translate from one function to
    a function with just sines and cosines. For
    example, you can always write tan, sec, csc, cot
    in terms of either sine or cosine.
  • Remember the following identities from
    PRE-CALCULUS!!!

16
IMPORTANT IDENTITIES
17
PROCEDURE
  • Well Im afraid to say it, but there is really
    no procedure or real template in attacking these
    problems except proper planning.
  • This takes a great deal of practice

18
EXAMPLE 1
  • Given
  • Best thing to do is to break the cosine function
    down to a 2nd degree multiplied by a 1st degree
    cosine.
  • Since cos2x1-sin2x, you can replace it.
  • Use u-substitution to solve the integral.

19
EXAMPLE2
  • Given
  • Break the 4th degree sine to two 2nd degre sines.
  • Use the sin2x theorem.
  • Expand the binomial squared.
  • For the cos22x, use the cos2x theorem.
  • Simplify
  • Dont forget the C!

20
EXAMPLE 3
  • Whenever you see a tan, sec, csc, or cot, always
    convert them to sines and cosines. This way, you
    can cancel or combine whenever necessary.
  • In this case, tan2xsin2x/cos2x. We are also
    lucky that the cos2x cancels.
  • Using the sin2x theorem, we can simply integrate.

21
NOTE ABOUT TRIGONOMETRIC INTEGRALS
  • There is no real rule for such integrals. But
    always remember
  • 1) If there is a mix of sines and cosines, break
    them up until they resemble an easier form
  • 2) Use any trig theorem that would be relevant to
    make a problem simpler.
  • 3) Convert everything to sines or cosines.

22
4) TRIGONOMETRIC SUBSTITUTION
  • Remember when we took derivatives of inverse
    trigonometric functions, we commonly dealt with
    sums or differences of squares.
  • Similarly, integrating sums of differences of
    square will lead us to the inverse trig
    functions.
  • However we need a stepping stone to integrate
    such functions.

23
GENERAL PROCEDURE
Given that a is a constant, and u is a function,
then follow the
IF YOU HAVE THIS..
THEN USE THIS FORMULA
24
THETA?
  • Since we are working with a substitution, theta
    would be the variable to use subsitution
  • Doesnt make sense? Lets do an example problem

25
EXAMPLE 1
  • Initially a very bad looking problem
  • Focus on the denominator, inside the radical, you
    have 9-x2. In effect, that is a2-u2, a being 3
    and u being x. If a2-u2 is used, then according
    to the table in the last slide, we would use ua
    sinq.
  • You already found a name for x. You need to give
    a name for dx. Differentiate x with respect to
    q. Solve for dx.

26
EXAMPLE 1
  • Here is the original problem
  • With the substitutions of x and dx, here is the
    original problem
  • Simplify a little
  • Pull out constants when needed.

27
EXAMPLE 1
  • The denominator is actually cos2x, according to
    the trig theorem. (Memorize them!)
  • Simplify
  • Integrate
  • We have our answer in terms of q!!! We need it in
    terms of x!

28
EXAMPLE 1
  • Dont forget what we said earlier. That x3 sin
    q. We need to know what q is in order to find out
    the solution in terms of x.

29
EXAMPLE 1
  • Simply replace all the q expressions with x
    expressions.
  • Simplify
  • Add constant!!
  • Sighs!! Were finished!!!

30
That was a LOT of work!!!
  • Here are trig substitution steps
  • 1) Find the correct equality statement using the
    table.
  • 2) Make the proper substitutions. Remember to
    have a substitute for x as well as dx.
  • 3) Integrate in terms of q.
  • 4) Convert all q terms to x terms.

31
5) RATIONAL FUNCTIONS
  • Of course, there will always be functions in the
    form of a ratio of two functions.
  • Two integrate most rational functions, the method
    of partial fractions come into play.
  • ltltBreak from Calculus entering Algebra
    Territorygtgt

32
PARTIAL FRACTIONS
  • This means you take a fraction and break it down
    into a sum of many fractions.
  • This way, we can add up the integrals of simpler
    easier fractions.

33
EXAMPLE
  • Given
  • The denominator could be factored to (x5)(x-2).
    This way we could have new denominators for the
    two new fractions.
  • Add these new fractions and distribute. Make sure
    you bring all x terms together, as well as
    bringing all the constants together.

34
EXAMPLE
  • The coefficient of x on the right side is 1. In
    order to keep the equality true, the coefficient
    of x on the left side should also equal 1.
  • AB1
  • Same thing with the constant. If the equality
    holds true, then -2A5B must equal -9.
  • To solve for A and B, you use methods from
    algebra. ltSystem of linear equationsgt.
  • If you multiply AB1 by 2, you will see that
    B-1. Therefore A2.

35
EXAMPLE
  • Since A2 and B-1, we can simply plug them in.
  • And integrate!!!
  • And the final answer!!

36
ANOTHER EXAMPLE
  • Given
  • Note If the numerator has a higher degree than
    the denominator, then do long polynomial
    division.
  • If you actually do the long division, you will
    get x-1-1/(x1). This is very easy to integrate.

37
POINTERS OF PARTIAL FRACTIONS
  • 1) Check if the top degree is bigger than the
    bottom. If so, perform long division
  • 2) If the denominator is factorable, then assume
    that the denominators of the new fractions will
    be those factors.

38
6) QUADRATIC DENOMINATOR PROBLEMS
  • This is really no different than trigonometric
    substitution.
  • Strictly rational functions with a quadratic
    denominator that cannot be reduced.
  • To make the denominator easier to work with, you
    must complete the square

39
COMPLETING THE SQUARE
  • Given problem
  • Look at the denominator. Take the coefficient of
    x and divide it by 2.
  • Take this result and square it.
  • 4/2 224
  • This result would form a perfect square when
    added to x24x.
  • The perfect square would be (x2)2.
  • However, we have a 5. If you add and subtract 4,
    combining 5 and -4 will yield 1.
  • You have a form of u2a2! Time for trig
    substitution!

40
EXAMPLE
  • Since we have u2a2, we must use the fact of
    uatan(q).
  • ux2 while a1
  • Substitute the values in appropriate spots.

41
EXAMPLE (work)
42
POINTERS
  • 1) Make the denominator into one of the three
    forms that allows trig substitution by the use of
    completing the square.
  • 2) Follow rules of trig substitution.

43
7) IMPROPER INTEGRALS
  • This is not an integral evaluating technique.
  • An improper integral is basically an integral
    that has infinity as its limits or has a
    discontinuity within its limits.

44
IMPROPER INTEGRALS
  • Examples of improper integrals

45
IMPROPER INTEGRALS
  • With limits of infinity, just use a letter to
    replace the infinity and treat as a limit.
  • And integrate as if nothing ever happened ?
  • Dont forget to use the limit.
  • Amazing! As we start from 0 to infinity, we get
    closer to 1 square unit of area! We say that it
    converges to 1.

46
IMPROPER INTEGRALS
  • Since we have a discontinuity in this function at
    x-2. To take this into account, we must split
    the integral into two parts. In addition, we
    cannot go exactly -2, but we have to get there
    pretty darn close. Therefore, we must use the
    one-sided limits from Chapter 1 to represent
    this. Two integrals one from -3 to a little
    before -2, and a little after -2 to 2.
  • Other than that, simply integrate ?
  • Notice how we got an answer that dont exist!!
    D.N.E (does not exist)! This means that this
    integral diverges. Also if an integral goes to
    infinity, it diverges.

47
POINTERS OF IMPROPER INTEGRALS
  • Remember to identify all the points of
    discontinuity. Remember to use limits before and
    after the points of discontinuity.
  • If you have infinity as your limit, remember to
    use infinity as your limit.
  • Other than that, use ALL of the previous
    techniques of integration mentioned.

48
FUNCTIONS WITHOUT AN ANTIDERIVATIVE
  • Besides three more chapters, this is the last of
    the single variable calculus. That is to say
    yf(x) in 2-dimensional x,y graph.
  • Before moving on, I must admit even though all
    continuous functions have derivatives, not all
    continuous functions have simple integrals in
    terms of elementary functions.
  • Elementary functions are adding, subtracting,
    multiplication, division, power, rooting,
    exponential, logarithmic, trigonometric, all of
    their inverses as well as combinations or
    composition functions. Basically, all the
    functions you ever used were composed of
    elementary functions.
  • Some functions do not have elementary
    antiderivatives. For example the classic (sin
    x)/x problem.
  • No matter what method you tried. Neither by
    u-substitution, integration by parts, trig
    substitution, partial fractions, or even guess
    and check will get you an antiderivative.
  • From my experience from differential equations
    class last year, the integral of sin(x)/x is
    Si(x) also known as the sine integral!
  • YOU DONT NEED TO KNOW THAT!!!!

49
OUTTA THIS WORLD FUNCTIONS!!!!!!!
  • You will be dealing with functions like erf(x),
    Si(x), Ci(x), Shi(x), Chi(x), FresnelS(x), and
    FresnelC(x). Take their derivatives and youll
    get regular sane functions. ? AAHHH!!! HARI
    BOL!!!!

50
SUMMARY
  • Actually, for once, looking at the length and
    material of this chapter. I am quite amazed to
    say that I have no words to summarize this
    chapter. There has been so many methods of
    integration. Namely u-substitution, integration
    by parts, how to deal with trig integrals, trig
    substitution, partial fractions, quadratic
    denominators, and improper integrals.
  • All I can say is that review this material over
    again!!!
  • Like I said previously, there is no set way to do
    these problems. There are more than one way of
    doing it.
  • You have to know what to do when which problem
    arrives at you.

51
CREDITS
  • Dr. A. Moslow
  • Dr. W. Menasco
  • Mr. G. Chomiak
  • Calculus and Early Transcendental Functions 5th
    Ed.
  • Finney Calculus
  • Single-Variable Calculus (SUNY Buffalo)
  • Princeton Review AP Calculus AB

52
NEED HELP?
  • Need help??
  • E-mail kksongs_1_at_hotmail.com
  • Please read help statement

53
END OF CHAPTER EIGHT
  • jaya sri krsna caitanya prabhu nityananda
  • sri advaita gadadhara sri vasadi gaura bhakta
    vrnda
  • hare krsna hare krsna krsna krsna hare hare
  • hare rama hare rama rama rama hare hare

54
END OF CHAPTER EIGHT
Write a Comment
User Comments (0)
About PowerShow.com