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Thread: Integration By Parts

  1. #1
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    Integration By Parts

    I am doing integration by parts and am struggling a little bit with two revision questions:

    The first is:

    1) \int cos(2x).(5x+1).dx

    The answer I have is - (5x+1)sin2x/2+5/4cos2x

    Is this right although I am not sure how I got there.

    The other question is a Definite Integral:

    2) \int (x+2)sinx.dx and the limits are 1 and 2

    I just can't seem to get the right indefinite integral to be able to replace x with the limits.

    The answer I get is (x+2)(-cos(x))+sin(x) but when O replace x with the upper and lower limit I get the wrong answer.

    Can anyone please help me.

    Thanks

    M
    Last edited by mm874; May 25th 2011 at 08:18 AM. Reason: Alter my answer
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  2. #2
    MHF Contributor
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    1) is correct. Will post a pic to show how.

    2) Are you subtracting cos x, there, or... should there be brackets...?

    1) Just in case a picture helps...



    ... where (key in spoiler) ...

    Spoiler:


    ... is the product rule. Straight continuous lines differentiate downwards (integrate up) with respect to x. And,



    ... is lazy integration by parts, doing without u and v.


    2) is much the same... you try.

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  3. #3
    MHF Contributor
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    OK, you've now put the brackets in... is that not working?

    Again, just in case...



    (And follow the key in post 2.)

    _________________________________________

    Don't integrate - balloontegrate!

    Balloon Calculus; standard integrals, derivatives and methods

    Balloon Calculus Drawing with LaTeX and Asymptote!
    Last edited by tom@ballooncalculus; May 25th 2011 at 08:39 AM. Reason: pic
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  4. #4
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    Thank you that makes sense. Are you also able to help me with the following as well?

    ∫(7-x)sin(x).dx

    I know the answer is:

    (x-7).cos(x)-sin(x)

    but i can't figure out why the (7-x) is swapped over and becomes (x-7).

    M
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  5. #5
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    (x - 7) = - (7 - x)

    The negative belonged originally to the cos (but then to the whole product). I.e. it's just a neat way of removing the negative.

    We could balloonify it as either...



    ... or...




    _________________________________________

    Don't integrate - balloontegrate!

    Balloon Calculus; standard integrals, derivatives and methods

    Balloon Calculus Drawing with LaTeX and Asymptote!
    Last edited by tom@ballooncalculus; May 25th 2011 at 01:29 PM. Reason: pics
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