1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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

    User Avatar
    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


    User Avatar
    Science Advisor
    Homework Helper
    Gold Member
    Dearly Missed

    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


    User Avatar
    Science Advisor
    Homework Helper
    Gold Member
    Dearly Missed

    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.
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?

Similar Discussions: Patterns in Integrals
  1. Number pattern (Replies: 5)

  2. Traffic Flow Pattern (Replies: 0)