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Math Help - integrals question

  1. #1
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    integrals question

    Hi i have having problems with a few integral questions. I am just totally lost of what to do.

    Let a > 0 and a not equal to 1. Find the indefinite integrals:



    ________________ + C



    ________________ + C

    and

    Evaluate the definite integral.



    ________________

    and

    Evaluate the indefinite integral



    ________________ + C


    and

    Find the indefinite integral



    ________________ + C


    I know these are alot of problems but I don't know how to solve for any of them.

    Thank you
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  2. #2
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    Some of those look ugly, but I can certainly help with the last one.

    (I'm just going to put INT for integration)

    INT ( x sin (2x) ) dx

    You can integrate this by parts.

    So let u = x and v'= sin 2x
    then u' = 1 and v = 0.5 cos 2x

    Then INT ( u v' ) dx = uv - INT ( v u' )

    => INT ( x sin (2x) ) dx = 0.5 x cos(2x) - 0.5 INT ( cos (2x) ) dx

    = 0.5 x cos(2x) + 0.25 sin 2x + C
    Last edited by TheBrain; February 15th 2007 at 03:18 PM.
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  3. #3
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    I can see how to do the second one as well though that will take more working...


    INT ( cosx [sinx]^a ) dx

    You can integrate this by parts.

    So let u = [sinx]^a and v' = cosx
    Then u' = a cosx [sinx]^a-1 and v = sinx

    Then INT ( u v' ) dx = uv - INT ( u' v )

    => INT ( cosx [sinx]^a ) dx = [sinx]^a+1 - a INT ( cosx [sinx]^a ) dx

    Now you can see this last term is just equal to what you started with times a.

    => {a+1} INT ( cosx [sinx]^a ) dx = [sinx]^a+1

    => INT ( cosx [sinx]^a ) dx = [ (sinx)^a+1 ] / [a+1] + C
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  4. #4
    Grand Panjandrum
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    Quote Originally Posted by killasnake View Post
    Hi i have having problems with a few integral questions. I am just totally lost of what to do.

    Let a > 0 and a not equal to 1. Find the indefinite integrals:



    ________________ + C
    int cos(x) a^(sin(x)) dx = int cos(x) e^[ln(a) sin(x)] dx

    Now you should recognise that ln(a) cos(x) is the derivative of ln(a) sin(x),
    so:

    int cos(x) e^[ln(a) sin(x)] dx = int d/dx{ (1/ln(a) e^[ln(a) sin(x)] } dx

    ..........................=1/ln(a) e^[ln(a) sin(x)] +C

    RonL
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  5. #5
    Forum Admin topsquark's Avatar
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    The methods below work, but I would suggest you do this first:
    Note that d/dx(sin(x)) = cos(x).

    So for the first two problems I would recommend putting
    u = sin(x)
    then look at the integrals and see what you can do with them.

    For the third problem, a similar approach will work:
    u = cos(x)

    -Dan
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