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!

Finding ∫x sin^3x dx

  1. Jan 25, 2012 #1
    1. The problem statement, all variables and given/known data

    Finding ∫x sin^3x dx

    2. Relevant equations

    I don't think this is needed

    3. The attempt at a solution
    ∫x sin^3x dx using integration by parts u = x, du = dx, dv = sin^3x, v = 1/3cos^3x - cosx
    = (x)(1/3cos^3x - cosx) - ∫1/3cos^3x - ∫cosx dx
    = (x)(1/3cos^3x - cosx) - 1/3∫cos^3x dx + (sinx)
    = (x)(1/3cos^3x - cosx) - 1/3∫cos^2xcosx dx + (sinx)
    = (x)(1/3cos^3x - cosx) - 1/3∫(1-sin^2x)(cosx) dx + (sinx)
    = (x)(1/3cos^3x - cosx) - (1/3∫cosx dx - ∫cosxsin^2x dx) + (sinx)
    = (x)(1/3cos^3x - cosx) - (1/3 (-sinx) - ∫cosxsin^2x dx) + (sinx)
    using subsitution u = sinx, du = cosx dx
    = (x)(1/3cos^3x - cosx) - (1/3 (-sinx) - ∫u^2 du) + (sinx)
    = (x)(1/3cos^3x - cosx) - (1/3 (-sinx) - 1/3u^3) + (sinx)
    then subsitute back
    = (x)(1/3cos^3x - cosx) - (1/3 (-sinx) - 1/3sin^3x) + (sinx) + C

    Anything I did wrong?
    Anyway to do this much easier/faster?
     
  2. jcsd
  3. Jan 25, 2012 #2
    Don't integrate by parts. Not yet, at least.

    Use the half-angle identity:
    sin2x = 1/2(1 - cos(2x))
     
  4. Jan 25, 2012 #3
    Ok, I do that then
    I'll get something like:

    = 1/2∫(1-cos2x)xsinx dx
    = 1/2∫xsinx - ∫xsinxcos2x dx <-- I get stuck here
     
  5. Jan 25, 2012 #4

    vela

    User Avatar
    Staff Emeritus
    Science Advisor
    Homework Helper
    Education Advisor

    You can always check your answer by differentiating it and seeing if you recover the integrand.
     
  6. Jan 25, 2012 #5

    SammyS

    User Avatar
    Staff Emeritus
    Science Advisor
    Homework Helper
    Gold Member

    To find v by integrating, I would rewrite sin3 x as

    sin3 x = (1 - cos2 x)sin  x
    = sin  x - sin  x   cos2 x​

    This is fairly easy to integrate.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Finding ∫x sin^3x dx
  1. Integral (x e^-3x dx) (Replies: 2)

Loading...