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Thread: Indefinite integrals

  1. #1
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    Exclamation Indefinite integrals

    Question:
    $\displaystyle \int(x^2-4x+4)^\frac{4}{3} . dx $

    i know that $\displaystyle x^2-4x+4 $ simplifies to $\displaystyle (x-2)^2 $
    So you can say the equation is:
    $\displaystyle \int (x-2)^\frac{8}{3} .dx $

    But i dont know how to solve it.
    Please answer in steps Thank you
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  2. #2
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    Quote Originally Posted by mj.alawami View Post
    Question:
    $\displaystyle \int(x^2-4x+4)^\frac{4}{3} . dx $

    i know that $\displaystyle x^2-4x+4 $ simplifies to $\displaystyle (x-2)^2 $
    So you can say the equation is:
    $\displaystyle \int (x-2)^\frac{8}{3} .dx $

    But i dont know how to solve it.
    Please answer in steps Thank you
    Let $\displaystyle u = x-2 \Rightarrow dx = du$.

    Then: $\displaystyle \int (x-2)^\frac{8}{3} ~dx = \int u^{\frac{8}{3}} ~ du$

    Can you solve it now?
    Last edited by Defunkt; Jan 9th 2010 at 06:24 AM. Reason: typo
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  3. #3
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    Quote Originally Posted by mj.alawami View Post
    Question:
    $\displaystyle \int(x^2-4x+4)^\frac{4}{3} . dx $

    i know that $\displaystyle x^2-4x+4 $ simplifies to $\displaystyle (x-2)^2 $
    So you can say the equation is:
    $\displaystyle \int (x-2)^\frac{8}{3} .dx $

    But i dont know how to solve it.
    Please answer in steps Thank you
    You are DEFINITELY on the right track.

    Now use a $\displaystyle u$ substitution.

    Let $\displaystyle u = x - 2$ so that $\displaystyle \frac{du}{dx} = 1$.

    The integral becomes

    $\displaystyle \int{u^{\frac{8}{3}}\,\frac{du}{dx}\,dx}$

    $\displaystyle = \int{u^{\frac{8}{3}}\,du}$

    $\displaystyle = \frac{3}{11}u^{\frac{11}{3}} + C$

    $\displaystyle = \frac{3}{11}(x - 2)^{\frac{11}{3}} + C$.
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  4. #4
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    Quote Originally Posted by Defunkt View Post
    Let $\displaystyle u = (x-2)^2 \Rightarrow dx = du$.

    Then: $\displaystyle \int (x-2)^\frac{8}{3} ~dx = \int u^{\frac{8}{3}} ~ du$

    Can you solve it now?
    Attempt:
    $\displaystyle \frac{3}{11} (x-2)^\frac{11}{3}+C $

    Is this the answer?
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  5. #5
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    Quote Originally Posted by mj.alawami View Post
    Attempt:
    $\displaystyle \frac{3}{11} (x-2)^\frac{11}{3}+C $

    Is this the answer?
    Yes, Well done!
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