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Integration by parts (problem plus question)

  1. Jan 26, 2013 #1
    1. The problem statement, all variables and given/known data
    I've run into this problem a few times, where I get the right answer, but multiplied by a constant where I would have it divided by the constant or vice versa.

    "First make a substitution and then use integration by parts to evaluate the integral"

    ∫cos(√x)dx


    2. Relevant equations

    ∫udv = uv - ∫vdu

    3. The attempt at a solution

    let A = √x
    dA = dx/(2√x)
    2(√x)dA = dx
    A2 = x

    2∫Acos(A)dA

    let u=A
    du=dA
    dv= cos(A)
    v = sin A + C

    2∫Acos(A)dA = Asin(A) - ∫sin(A)
    = Asin(A) + cos(A)
    so, then

    ∫Acos(A)dA = (Asin(A) +cos(A))/2 +C

    this is wrong, it should be 2Asin(A) + 2cos(A) + C and im not sure where exactly I can remedy this.

    I think my problem might be with 2∫Acos(A)dA = Asin(A) - ∫sin(A)
    since I have an integral im evaluating by parts multiplied by a constant, does
    2∫Acos(A)dA = Asin(A) - ∫sin(A) => 2∫Acos(A)dA = 2(Asin(A) - ∫sin(A)) ???

    or more generally c∫udv = c(uv - ∫vdu) ?
     
  2. jcsd
  3. Jan 26, 2013 #2

    Curious3141

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    Homework Helper


    You missed out the "dA" there at the end. But most importantly, you missed out the factor of 2 on the right hand side.

    $$\int A\cos A dA = A\sin A - \int \sin A dA$$

    and when you multiply by 2, you should do it throughout.

    What?! Why? Something is wrong with your algebra. See what I wrote above.

    Yes, these two statements are correct (except you've missed out a "dA" again).

    You don't have to bother with constants of integration at all until the final answer step. Even writing dv = cos A, hence v = sin A + c is not necessary.

    You should express everything in terms of the original variable (x) in the final answer.
     
  4. Jan 26, 2013 #3
    Okay thank you. My problem was not distributing the constant from
    2∫Acos(A)dA = Asin(A) - ∫sin(A)dA
    this should be
    2∫Acos(A)dA=2( Asin(A) - ∫sin(A)dA)
    and in terms of x:

    =2(√x)(sin(√x) + 2cos(√x )

    yes.
    Thanks bunches :approve:
     
  5. Jan 26, 2013 #4

    Curious3141

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    Homework Helper

    Your brackets are off - it should be ##2(\sqrt x \sin \sqrt x + \cos \sqrt x) + c##. Be careful about things like this, and don't forget your constant at the end. :smile:
     
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