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Math Help - Need Help With Indefinite Integral

  1. #1
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    Need Help With Indefinite Integral

    This particular integral is driving me crazy. I can't figure it out for the life of me. Could someone please explain to me how to go about solving it?

    Integrate[5x(7x - 9)^14dx

    I was given a few hints but they still are not helping me at all. First let U=7x-9 and then let x=(U+9)/7. I would imagine this should be done using the substitution method. Thanks in advance

    John
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  2. #2
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    Quote Originally Posted by TreeMoney
    This particular integral is driving me crazy. I can't figure it out for the life of me. Could someone please explain to me how to go about solving it?

    Integrate[5x(7x - 9)^14dx

    I was given a few hints but they still are not helping me at all. First let U=7x-9 and then let x=(U+9)/7. I would imagine this should be done using the substitution method. Thanks in advance

    John
    Is that
    INT.[5x (7x -9)^14]dx ?

    If yes, then here is one way of doing it.

    Integration by parts:
    INT.[u]dv = u*v -INT.[v]du ------***

    Let u = 5x
    So, du = 5dx

    And dv = (7x -9)^14 dx = (7x-9)^14 *(7dx /7) = (1/7)(7x-9)^14 *(7dx)
    So, v = (1/7)[(1/15)(7x-9)^15] = (1/105)(7x-9)^15

    Hence,
    INT.[5x (7x-9)^14]dx
    = (5x)(1/105)(7x-9)^15 -INT.[(1/105)(7x-9)^15](5dx)
    = (1/21)x(7x-9)^15 -(1/21)INT[(7x-9)^15]dx
    = (1/21)x(7x-9)^15 -(1/21){(1/7)INT[(7x-9)^15](7dx)}
    = (1/21)x(7x-9)^15 -(1/147){(1/16)(7x-9)^16}
    = (1/21)x(7x-9)^15 -(1/2352)(7x-9)^16 -------------answer.

    Check,
    d/dx {(1/21)x(7x-9)^15 -(1/2352)(7x-9)^16}
    = (1/21){[x*15(7x-9)^14 *7] +[(7x-9)^15 *1]} -(1/2352){16(7x-9)^15 *7}
    = {5x(7x-9)^14 +(1/21)(7x-9)^15} -(1/21)(7x-9)^15
    = 5x(7x-9)^14
    Okay.
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  3. #3
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by TreeMoney
    This particular integral is driving me crazy. I can't figure it out for the life of me. Could someone please explain to me how to go about solving it?

    Integrate[5x(7x - 9)^14dx

    I was given a few hints but they still are not helping me at all. First let U=7x-9 and then let x=(U+9)/7. I would imagine this should be done using the substitution method. Thanks in advance

    John
    It can be done by integration by parts, but also directly by substitution.
    \int dx \, 5x(7x-9)^{14} = \int \left ( \frac{du}{7} \right ) \, 5 \left ( \frac{u+9}{7} \right ) u^{14}
    =\frac{5}{7} \int du \, \left ( \frac{u}{7} + \frac{9}{7} \right )u^{14}
    =\frac{5}{7} \int du \, \frac{1}{7}u^{15} + \frac{5}{7} \int du \, \frac{9}{7}u^{14}
    =\frac{5}{7} \frac{1}{7}\frac{1}{16}u^{16} + \frac{5}{7} \frac{9}{7}\frac{1}{15}u^{15}
    =\frac{5}{784}(7x-9)^{16}+\frac{3}{49}(7x-9)^{15}

    Which you can then simplify any way you wish.

    -Dan
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  4. #4
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    Thanks Guys. Now that I see how it is done, I was making stupid algebra mistakes. Thanks again for the help. It's much appreciated.

    John
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  5. #5
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by TreeMoney
    Thanks Guys. Now that I see how it is done, I was making stupid algebra mistakes. Thanks again for the help. It's much appreciated.

    John
    There's no such thing as a "stupid mistake." You either get it right or you don't. You didn't get it right, you knew you didn't, and you asked a question about it. I see no stupidity there, I see a good practice in action!

    -Dan
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