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Reduction Formula

  1. Jun 9, 2006 #1
    Hi, Im having trouble understanding this question, I have looked over a few examples, but I'm still confused about the process.

    A)Use the reduction formula to show that:

    [tex] \int sin^2xdx = \frac {x}{2} - \frac{sin2x}{4} + C[/tex]

    any help would be appreciated
  2. jcsd
  3. Jun 9, 2006 #2


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    I assume you're referring to the reduction of the exponent?

    Using cos(2x) = cos²x-sin²x combined with cos²x+sin²x = 1, you can derive the following formulas to get rid of a square in cos or sin:

    sin²x = (1-cos(2x))/2 and cos²x = (1+cos(2x))/2

    Try to verify this yourself.

    Now, using the first formula, do you see how the integral was done?
  4. Jun 9, 2006 #3
    i am still confused, this is the first question like this I have done. The question says to refer to ex.6...here it is:

    [tex] \int sin^nxdx = -\frac{1}{n}cosxsin^n^-^1x + \frac{n-1}{n} \int sin^n^-^2xdx[/tex]

    let: [tex]u=sin^n^-^1[/tex]


    [tex]du = (n-1)sin^n^-^xcosxdx

    [tex] v=-cosx[/tex]

    integration by parts;

    [tex]\int sin^nxdx = -cosxsin^n^-^1x + (n-1) \int sin^n^-^2xcos^2xdx[/tex]
  5. Jun 9, 2006 #4


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    I see, they really mean a reduction formula for the integral (a bit overkill for such an integral, imho).

    In that case, compare the formula (your first line) with the problem. It's exactly the same, only n = 2.
    So apply the formule with n = 2, no integration by parts is necessary (unless you'd want to prove the reduction formula, but that isn't asked here!)
  6. Jun 9, 2006 #5
    [tex] \int sin^2xdx = \frac {x}{2} - \frac{sin2x}{4} + C[/tex]

    so [tex] u=sin^n^-^1[/tex]...i get that part..and end up with only -sinx
  7. Jun 9, 2006 #6


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    Are you trying to prove the reduction formula you gave?
    I don't understand why you keep coming that this 'u' for a substitution.

    I understand the problem as:


    [tex] \int sin^2xdx [/tex]

    Using the formula

    [tex] \int sin^nxdx = -\frac{1}{n}cosxsin^n^-^1x + \frac{n-1}{n} \int sin^n^-^2xdx[/tex]

    Is that what you're supposed to do? If so, apply this last formula with n = 2.
  8. Jun 9, 2006 #7
    I am trying to show[tex] \int sin^2xdx = \frac {x}{2} - \frac{sin2x}{4} + C[/tex]
    using the reduction formula shown in example 6:[tex] \int sin^nxdx = -\frac{1}{n}cosxsin^n^-^1x + \frac{n-1}{n} \int sin^n^-^2xdx[/tex]
  9. Jun 9, 2006 #8
    and in that example they let [tex]u=sin^n^-^1[/tex]etc...shouldnt i do the same for what i am trying to show?
  10. Jun 9, 2006 #9
    i made this too complicated...hahaha so easy..nvm i understand now

  11. Jun 9, 2006 #10


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    I think that in example 6, they have proven this formula. In order to do this, they'll have used integration by parts I assume.
    What you now have to do (*I think*), is use this formula (not prove it again) on the particular problem.

    In ex 6, they've set up a relation between the integral of sin(x)^n and an integral with sin(x)^(n-2), so this formula allows you to reduce the exponent by 2 every time you apply it. Now in your problem, you wish to find the primitive of sin²x, you can use this formula with n = 2.
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