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Math Help - integration using generalized power rule

  1. #1
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    integration using generalized power rule

    integral(x^5*(sqrt(13+x^6))

    Here's what I did... but its apparently wrong

    1. Realized I needed to use the generalized power rule
    2. set x^6+13=u
    3. set 6*x^5=du
    4. solved for the coeffecient. x^5*Z=du Z=6

    so now I have

    6*int(u^(1/2))*du

    from there i got

    (6*u^(3/2))/(3/2)

    i plugged in the u...

    (6*(x^6+13)^(3/2))/(3/2)

    but this is incorrect... what'd i do wrong?
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  2. #2
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    Hello, thedoge!

    \int x^5\sqrt{13+x^6}\,dx

    Let u = 13 + x^6\quad\Rightarrow\quad du = 6x^5dx\quad\Rightarrow\quad dx = \frac{du}{6x^5}

    Substitute: . \int x^5\cdot u^{\frac{1}{2}} \cdot\frac{du}{6x^5} \;=\;\frac{1}{6}\int u^{\frac{1}{2}}du \;=\;\frac{1}{6}\cdot\frac{2}{3}u^{\frac{3}{2}} + C \;=\;\frac{1}{9}u^{\frac{3}{2}} + C

    Back-substitute: . \frac{1}{9}(13 + x^6)^{\frac{3}{2}} + C

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