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Math Help - evaluate the indefinite integral

  1. #1
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    evaluate the indefinite integral

    Evaluate the indefinite integral of x(2x+5)^8dx

    Alright, so far I've got:

    u=2x+5, du=2dx, (1/2)du=dx, x = (u-5)/2
    integral of (u-5)/2*u^8*1/2du
    1/2integral of (u-5)/2*u^8du
    1/2 ((2x+5)-5)/2*((2x+5)^9)/9 + C

    But this wasn't the answer...8( Am I doing something wrong?
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  2. #2
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    You would need to use integration by parts...

    Let \displaystyle u = x and \displaystyle dv = (2x + 5)^8.

    Then \displaystyle du = 1 and \displaystyle v = \frac{1}{18}(2x + 5)^9.


    So \displaystyle \int{x(2x+5)^8\,dx} = \frac{1}{18}x(2x + 5)^9 - \int{\frac{1}{18}(2x + 5)^9\,dx}

    \displaystyle = \frac{1}{18}x(2x + 5)^9 - \frac{1}{36}\int{2(2x + 5)^9\,dx}

    \displaystyle = \frac{1}{18}x(2x + 5)^9 - \frac{1}{360}(2x + 5)^{10} + C.
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  3. #3
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    Quote Originally Posted by Taurus3 View Post
    Evaluate the indefinite integral of x(2x+5)^8dx

    Alright, so far I've got:

    u=2x+5, du=2dx, (1/2)du=dx, x = (u-5)/2
    integral of (u-5)/2*u^8*1/2du
    1/2integral of (u-5)/2*u^8du
    1/2 ((2x+5)-5)/2*((2x+5)^9)/9 + C

    But this wasn't the answer...8( Am I doing something wrong?

    \frac{1}{4}\int(u-5)u^8du=\frac{1}{4}\int(u^9-5u^8)du=\frac{1}{4}\left(\frac{u^{10}}{10}-\frac{5u^9}{9}\right)+C\,,\,\,C= constant
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  4. #4
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    however, the answer is 1/40(2x+5)^10-5/36(2x+5)^9+C
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  5. #5
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    Which is what you get after you back substitute u = 2x + 5.

    My answer is also correct, just written in a different form.
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  6. #6
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    Quote Originally Posted by Taurus3 View Post
    however, the answer is 1/40(2x+5)^10-5/36(2x+5)^9+C
    I'm assuming it is: \frac{1}{40}(2x+5)^{10} - \frac{5}{36}(2x+5)^9+C

    That is the same thing as...

    \frac{1}{4}\left(\frac{u^{10}}{10}-\frac{5u^9}{9}\right)+C

    Substitute back in u = 2x+5 and you will get the same answer.
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