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Definite integral problem

  1. Nov 20, 2013 #1
    1. The problem statement, all variables and given/known data
    Let ##\displaystyle f(r)=\int_0^{\pi/2} x^r\sin x \,\, dx##. Now match the following List-I with List-II.

    $$
    \begin{array} {|c| c | l c|}
    \hline
    & \text{List-I} & & \text{List-II} & & \\
    \hline
    \text{(P)} & \lim_{r\rightarrow \infty} r\left(\frac{2}{\pi}\right)^{r+1}f(r) & 1\cdot & 0 & & \\
    \\
    \\
    \\
    \text{(Q)} & \lim_{r\rightarrow \infty} \frac{f(r)}{f(r+1)} & 2\cdot & 1 & &\\
    \\
    \\
    \\
    \text{(R)} & \lim_{r\rightarrow \infty} \left(\frac{f(r)}{r\int_0^{\pi/2} x^r\cos x \,\, dx}\right) & 3\cdot & \frac{2}{\pi} & & \\
    \\
    \\
    \\
    \text{(S)} & \lim_{r\rightarrow \infty} \int_0^1 x^r\sin x \,\, dx & 4\cdot & \frac{\pi}{2} & &\\
    \hline
    \end{array}
    $$

    2. Relevant equations



    3. The attempt at a solution
    I haven't been able to make any useful attempt on this problem. I tried integrating by parts and reached the following:

    $$f(r)=r\left(\frac{\pi}{2}\right)^{r-1}-r(r-1)f(r-2)$$
    I am not sure if above is of any help.

    Any help is appreciated. Thanks!
     
    Last edited: Nov 20, 2013
  2. jcsd
  3. Nov 20, 2013 #2

    jedishrfu

    Staff: Mentor

    In your f(r) function integral aren't you missing argument for the sin function?
     
  4. Nov 20, 2013 #3
    Woops, edited! Thanks. :)
     
  5. Nov 20, 2013 #4

    jedishrfu

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    It seems that doing an integration may be too time consuming.

    Have you looked at the questions and the potential answers.

    For example, the last one where r goes to infinity with x ranging of 0 to 1, how would that curve look if you drew it for r=1 the r=2...?

    then which answer would most likely be its answer?
     
  6. Nov 20, 2013 #5
    Can we please look for a proper method? :rolleyes:

    I usually sketch the curve when everything else fails. I doubt that there is a need to sketch the curve in this case.
     
  7. Nov 20, 2013 #6

    jedishrfu

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    Okay, this looks like a speed exam problem where you may need to improvise. Sometimes they will give you a really tough integral to put you off track wasting time trying to evaluate it. I agree that if you had the integral evaluated the answers would probably pop right out.

    Okay so when I integrate by parts I define u=x^r and v'=sinx so that v=-cosx

    and get integral f(r) = [-x^r*cos(x) + x^(r+1)*cos(x)*1/(r+1) ] 0 to pi/2

    maybe by leaving it in that form the limits will be easier to evaluate.
     
    Last edited: Nov 20, 2013
  8. Nov 20, 2013 #7
    Yes, its an exam problem.

    Before I proceed with your approach, I was wondering if you could comment on my thoughts about this problem.

    In the first post, I found f(r) to be:
    $$f(r)=r\left(\frac{\pi}{2}\right)^{r-1}-r(r-1)f(r-2)$$
    I am thinking that if ##r\rightarrow \infty##, I can take ##f(r) \approx f(r-2)##, would this be valid?
     
  9. Nov 20, 2013 #8

    jedishrfu

    Staff: Mentor

    It doesn't seem right.

    Wouldn't the pi/2 term be bigger than f(r-2) term?

    You have an r factor that can be removed to make your relation more like:

    f(r) ~ r*f(r-2)

    How did you get your relation of f(r) = f(r-2) ?

    We need a mathematician here...
     
  10. Nov 21, 2013 #9
    My logic was that a function should have almost the same values near infinity. We have ##f(\infty)## and ##f(\infty-2)##, both should have almost the same value as ##r\rightarrow \infty##.
     
  11. Nov 21, 2013 #10

    jedishrfu

    Staff: Mentor

    Maybe you can send a PM to Mark44 or HallsofIvy who more versed in mathematics to look at your thread.

    If there is a method to solving this kind of problem they would know it.

    I am more of an amateur trying to relearn stuff I learned decades ago.
     
  12. Nov 21, 2013 #11

    Mark44

    Staff: Mentor

    I don't see anything wrong with sketching a curve.
    Sketching a graph is usually the first thing I do, not something that I try as a last resort. Getting a feel for the geometry of a problem brings in a whole different area of your brain than when you limit yourself to the area that works in symbols. In effect, you are tying one hand behind your back.
    I've just started looking at this problem, so I don't have much insight into it. However, the "Q" item in the first column is looking at the ratio of f(r) to f(r+1). I think you can rule out 0 as a potential answer.

    As far as the integral itself, integration by parts is definitely something that I would try - maybe twice.
     
  13. Nov 21, 2013 #12

    vela

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    This would probably be valid if f(r) converged to a finite value, but it doesn't appear to.

    I'd go with the integral in S being equal to 0. Just sketch ##x^r## for increasing powers of ##r## on the interval [0,1] to see this.
     
  14. Nov 21, 2013 #13

    vela

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    Since you know the limit in P is finite, you can deduce how f(r) behaves for large r. From that, you can get the likely answer for Q is #3.
     
  15. Nov 21, 2013 #14

    haruspex

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    S is easy. Just use 0 <= sin(x) <= sin(1) < 1 and 0 <= x <= 1.
    By elimination, that tells you P converges to a nonzero value. As vela suggests, it's then easy to find Q - just let c > 0 be the answer to P. That just leaves two choices for P and R. maybe it's easy to see which is larger?

    Fwiw, I think I solved the recurrence relation for f. Something like ##\frac 1{2i}s^{-r}((1+is)^r - (1-is)^{r})##, where ##s = \frac 2 \pi##
     
    Last edited: Nov 22, 2013
  16. Nov 22, 2013 #15
    Oh yes, that was easy. Thanks! :)
    What is c? :uhh:

    How? :confused:

    Can you please post the steps to solve the recurrence relation? It's not going to help me at the moment but I will someday return to this thread when I am done with recurrence relations. Thanks!
     
  17. Nov 22, 2013 #16

    pasmith

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    You need [itex]f(0) = \int_0^{\pi/2} \sin x\,dx = \cos 0 - \cos \frac\pi 2 = 1[/itex]. Your formula gives [itex]f(0) = 0[/itex].

    I agree that [itex]f[/itex] is the imaginary part of
    [tex]I(r) = \int_0^{\pi/2} x^r e^{ix}\,dx = \left( \frac \pi 2 \right)^r + irI(r-1)[/tex]
    subject to [itex]I(0) = 1 + i[/itex] but I don't think you've solved the recurrence correctly.
     
  18. Nov 22, 2013 #17

    vela

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    Is there perhaps a typo in one of the limits? I find both Q and R converge to the same value.
     
  19. Nov 22, 2013 #18
    No there is no typo but according to the answer key, both Q and R share the same answer.

    Any hints on how you determine that?
     
  20. Nov 22, 2013 #19

    haruspex

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    Any of answers 2, 3, 4 - it doesn't matter. Since S=0, P cannot be 0. So we know P converges to some constant c, 0 < c < ∞, giving us an expression for the asymptotic behaviour of f. Substitute that in Q.
    That was just a possibility, but I see another way.
    Go back and look at your integration by parts. You did two steps of that to get your recurrence relation, right? The first step gave you something very close to the denominator in R. Can you see whether that makes R more or less than 1?

    OK, but not right now. I did say 'something like'; I knew I had not started it off correctly at f(0), as pasmith notes.
     
  21. Nov 22, 2013 #20

    Ray Vickson

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    I don't think it is this simple. For integer r the recursions for even r and for odd r separate. For even r, the recursion for ##g(m) = f(2m)## is
    [tex] g(m) = \frac{2}{c} p^m - 2m (2m-1) g(m-1), \; g(0) = 1,[/tex]
    where ##c = \pi/2## and ##p = c^2 = \pi^2/4##. Maple gets the solution
    [tex] g(m) = \frac{(-1)^m 4^m}{\sqrt{\pi}} m! (m \, - \, 1/2)!
    \left( 1 - \frac{\sqrt{\pi}}{2c} \sum_{j=0}^{m-1} (-1)^j
    \frac{(j+1)p^{j+1}}{4^j (j+1)! (j \,+\,1/2)!} \right) [/tex]
    where ##u! \equiv \Gamma(u+1)## for any ##u##.

    We can get something similar for odd r.
     
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