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Patterns in Integrals

  1. Dec 1, 2007 #1
    1. The problem statement, all variables and given/known data
    Hi, im doing this discovery project called patterns in integrals i found in my calculus text book. I have to use a CAS (I'm using Maple) to investigate indefinite integrals of families of functions. Then by observing the patterns that occur in the integrals, i have to first guess, and then prove, a general formula for the integral of any member of the family. there are four different familes and im done with three of them, but stuck on the last one. I would appreciate any help. The question and what I have done so far is on the pdf attachment.

    3. The attempt at a solution

    My attempt is on the pdf file i attached.

    Attached Files:

  2. jcsd
  3. Dec 1, 2007 #2

    Gib Z

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    Homework Helper

    Please post the actual family here, since we have to wait for a mentor to approve the file first. I'm sure we can help you though, so thats some reassurance for you =] Welcome to Physicsforums!
  4. Dec 1, 2007 #3
    (a) use a CAS to evaluate the following integrals (I used maple)

    [tex]\int{xe^{x}}dx = \left( x-1 \right) {e^{x}}[/tex]
    [tex]\int{x^{2}e^{x}}dx = \left( 2-2\,x+{x}^{2} \right) {e^{x}}[/tex]
    [tex]\int{x^{3}e^{x}}dx = \left( -6+6\,x-3\,{x}^{2}+{x}^{3} \right) {e^{x}}[/tex]
    [tex]\int{x^{4}e^{x}}dx = \left( 24-24\,x+12\,{x}^{2}-4\,{x}^{3}+{x}^{4} \right) {e^{x}}[/tex]
    [tex]\int{x^{5}e^{x}}dx = \left( -120+120\,x-60\,{x}^{2}+20\,{x}^{3}-5\,{x}^{4}+{x}^{5}
    \right) {e^{x}}

    (b) based on the patterns of your responses in part (a), guess the value of [tex]\int{x^{6}e^{x}}dx[/tex] Then use your CAS to check your answer.

    This was my guess: [tex]e^{x}(x^{6}-6x^{5}+30x^{4}-120x^{3}+360x^{2}-720x+720)[/tex] and maple returned the same answer.

    (c) based on the pattern in parts (a) and (b), make a conjecture as to the value of the integral
    when n is a positive integer

    This is what i came up with: [tex]\sum_{i=0}^{n}\frac{x!}{i!}n!e^{x}
    Now this is where im stuck because i know this is not correct.
    I figured it has something to do with factorial or series.

    (d) use mathematical induction to prove the conjecture you made in part (c)
    Last edited: Dec 1, 2007
  5. Dec 1, 2007 #4


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    Hmm..not quite sure you got that right, here's a better approach:
    Define a sequence as follows:
    Thus, we have:

    Assume a solution as follows:
    Thus, inserting in our difference equation, we get:

    Therefore, we get:
    Last edited: Dec 1, 2007
  6. Dec 1, 2007 #5
    I appreciate your help arildno. thank you!
  7. Dec 2, 2007 #6
    I have one more question. Im a little loss on how to use mathematical induction to prove this, can u help me. thank you.
  8. Dec 3, 2007 #7


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    The mathematical induction step is taken care of by setting up the difference equation, valid for all n
  9. Dec 3, 2007 #8
  10. Dec 5, 2007 #9
    ive tried but no luck, im not good with mathematical induction.
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