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Thread: Indefinite Integration Using Substitution

  1. #1
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    Indefinite Integration Using Substitution

    Find $\displaystyle \int \frac{(lnx)^7}{x} dx$.

    Let $\displaystyle u = lnx$

    Let $\displaystyle du = \frac{1}{x} dx => dx = x du$

    $\displaystyle = \int \frac{1}{x} (lnx)^7 dx$

    $\displaystyle = \int (\frac{1}{x}) ((lnx)^7) (x) (du)$

    $\displaystyle = \frac{u^8}{8} + c$

    ... I stopped right here because I'm not getting the right answer... which is...

    $\displaystyle = \frac{1}{4}u^4 + c$

    I don't get where $\displaystyle \frac{1}{4}$ and $\displaystyle u^4$ comes from??? I thought I'm following the antiderivative rules???
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  2. #2
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    Quote Originally Posted by Macleef View Post
    Find $\displaystyle \int \frac{(lnx)^7}{x} dx$.

    Let $\displaystyle u = lnx$

    Let $\displaystyle du = \frac{1}{x} dx => dx = x du$

    $\displaystyle = \int \frac{1}{x} (lnx)^7 dx$

    $\displaystyle = \int (\frac{1}{x}) ((lnx)^7) (x) (du)$

    $\displaystyle = \frac{u^8}{8} + c$

    ... I stopped right here because I'm not getting the right answer... which is...

    $\displaystyle = \frac{1}{4}u^4 + c$

    I don't get where $\displaystyle \frac{1}{4}$ and $\displaystyle u^4$ comes from??? I thought I'm following the antiderivative rules???
    0.25u^4+C is wrong, your solution is absolutely correct.
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  3. #3
    is up to his old tricks again! Jhevon's Avatar
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    don't forget to back-substitute, the solution is $\displaystyle \frac 18 ( \ln x)^8 + C$
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