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Integration Bee

  1. Mar 18, 2005 #1
    And, yes, I mean what you think it means: antidifferentiating. Finding a primative. Whatever...

    At my school, we're conducting an integration bee not unlike similar bees done elsewhere.

    The purpose of this thread is two-fold:
    1. To compile a list of non-routine integrals
    2. To discuss how these integrals are done

    Here is the list I came up with. Please add to the list. I'm not looking for 300 integrals no one at a community college can solve (and, yes, by "solve", I mean to antidifferentiate). They should stump a significant percentage of those who would get an A in Integral Calculus though not 100%.

    A few of these are downright easy but they can stump the woefully inexperienced.

    One or two of them are potentially very difficult if you're not clever enough.

    Thanks for your input!
     
  2. jcsd
  3. Mar 18, 2005 #2
    What is the most difficult one according to you ?

    i will solve it

    marlon
     
  4. Mar 18, 2005 #3
    Most are standard textbooks problems.
     
  5. Mar 18, 2005 #4
    I would guess that #9 would stump the most people but I really have no idea.

    You may find #10 and #2 fun as well.
     
  6. Mar 18, 2005 #5
    Those are nice, but the tricks don't really get fun until definite integration:

    [tex] \int_{-\infty}^{\infty} e^{-x^2} dx = \sqrt{\pi} [/tex]

    [tex] \int_{-\infty}^{\infty}\frac{sin(x)}{x} dx = \pi [/tex]

    The first one is a fairly trivial change of variable trick, and the second is found by contour integration. In addition to having these down pat, an integrator champ should be able to pull all the tricks: differentiating under the integral sign, direct integration of differential eq corresponding to the function, etc.
     
    Last edited: Mar 18, 2005
  7. Mar 18, 2005 #6
    9 is probably the most easy one

    2 is easy with a good substitution and ten is nice

    marlon
     
  8. Mar 18, 2005 #7

    Tom Mattson

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    I like these:

    [tex]
    \int e^{ax} sin(bx)dx
    [/tex]

    [tex]
    \int e^{ax} cos(bx)dx
    [/tex]

    [tex]
    \int sec^n(x) dx
    [/tex]

    (n is odd and positive in the last one).

    They all have one thing in common: when integrating by parts, you have to recognize that you eventually get the same integral you started with, and you have to add a multiple of it to both sides to finish the problem.
     
    Last edited: Apr 1, 2005
  9. Mar 18, 2005 #8
    You think 9 is easier than 4, 5, 14, 15, and 19? Hmm... Is there an easy way to do #9 that I don't know about? What's the easy way to solve #9?
     
  10. Mar 18, 2005 #9
    9 is equal to

    [tex]\int \frac{dx}{(x^2 + 8)^2 -16 x^2}[/tex]

    then use a²-b² = (a-b)(a+b) in the denominator
    then integration by partial fractions

    marlon
     
  11. Mar 18, 2005 #10
    9 is easy because it is quite straightforeward this is the easiest way out

    marlon, though i admit it requires some calculation
     
  12. Mar 18, 2005 #11
    Yeah that's how I'd do it but I wouldn't call factoring x^4+64 "easy." Sure, compared to the proof of FLT it's easy but... ;)
     
  13. Mar 18, 2005 #12
    how did you do 10 ?

    marlon
     
  14. Mar 18, 2005 #13
    i got an answer by using the substitution t²=tanx and then apply partial fractions

    marlon
     
  15. Mar 18, 2005 #14
    I'm no integral bee champion so this is probably not the best way to do it:

    u=Tan[x] turns it into
    Sqrt/(1+u^2).

    Re-write:
    u/(Sqrt(1+u^2)).

    Now let v=Sqrt to get
    2v^2 / (1+v^4) which is integrable by partial fractions as the denominator splits into (v^2+Sqrt[2]v+1)(v^2-Sqrt[2]v+1).
     
  16. Mar 18, 2005 #15
    any other questions ?

    marlon
     
  17. Mar 21, 2005 #16

    dextercioby

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    [tex] \int e^{tan x} \ dx =... ? [/tex]

    [tex] \int \sqrt{\sin x} \ dx =... ? [/tex] Piece of cake for champions like Marlon
     
  18. Mar 21, 2005 #17
  19. Mar 21, 2005 #18
    The integral exp(tanx) is not that difficult, if i am right. Write the exp(tanx) as a series :
    sum over n of ((tanx)^n)/n!

    Now the integral sign and summation sign can be interchanged so what we really need to integrate is (tan(x))^n...

    i suppose that by writing tan as sin/cos, you can construct a recursion relation. This relation can be constructed by integration by parts...

    marlon
     
  20. Mar 21, 2005 #19
    holy... geez talk about scaring a Calc 1 student... I don't even know how to approach some of those integrals... geez. The only thing that comes close that I can solve is probably applying an arctangent rule to no. 9 from my knowledge, but as marlon said I probably have to review partial fractions.

    ack.
     
  21. Mar 21, 2005 #20

    dextercioby

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    Marlon,the [itex]\int e^{\tan x} \ dx [/itex] admits an exact (nonseries) solution among special functions.

    Daniel.
     
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