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Integrate arccos x by parts

  1. Sep 28, 2006 #1
    Could som1 plz help me integrate arccos x by parts. i've done examples using integration by parts but they were all some form of multiplication, ie
    y = xe^x, y = x sin x etc. i'm really unsure where to start with this problem :confused:
  2. jcsd
  3. Sep 28, 2006 #2
    Try a substitution like [itex]y=\arccos(x)[/itex], [itex]dx=\ldots[/itex] ?
    This should give you something you can integrate by parts.
  4. Sep 28, 2006 #3
    If i use a substitution as you suggested i get end up with two variables in my equation:

    Integral (arccos(x) dx)

    let y = arccos(x)

    dy/dx = -1/(sqrt(1-x^2)

    dx = -sqrt(1-x^2) dy

    Substitute into original equation:

    Integral (arccos(x) dx)

    => Integral (y * -sqrt(1-x^2)) dy <-----I have a y and and x in the equation :S what do i do now :S
    Last edited: Sep 28, 2006
  5. Sep 28, 2006 #4


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    I = Int ( Arccosx) dx
    I = xArccosx - x/Sqrt( 1- x^2) dx
    I = xArccosx - 1/2 * 2 Sqrt(1-x^2) + C
    I = xArccosx + Sqrt(1 - x^2) + C
  6. Sep 29, 2006 #5


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    Staff Emeritus
    Science Advisor

    Yes, in any substitution, it is your responsibility to see that you replace every instance of the old variable with a new. If y= arccos x then x= cos y so dx= - sin y dy.
    [tex]\int arccos x dx= -\int y sin y dy[/tex]
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