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Antiderivative help please.

  1. Oct 20, 2006 #1
    Hi, I am trying to integrate [tex]\int_{1}^{2} \frac{x^{2}+1}{\sqrt{x}}[/tex] using the Evaluation Theorem.

    So my first step is to find the antiderivative of [tex]\frac{x^{2}+1}{\sqrt{x}}[/tex].. And that is where my troubles lie.

    I start by rewriting it as [tex](x^{2}+1)*(x^{-1/2}}[/tex] but then realize that I don't know how to find the antiderivative..

    I tried using the rule [tex]x^{n}=\frac{x^{n+1}}{n+1} [/tex]

    and got [tex] (\frac{x^{3}}{3}+x)*2*\sqrt{x}[/tex] but this does not differentiate into the original function, can someone help me out?
     
    Last edited: Oct 20, 2006
  2. jcsd
  3. Oct 20, 2006 #2

    Hootenanny

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    Consider your function;

    [tex](x^2 +1)\cdot x^{-\frac{1}{2}}[/tex]

    Now open the parentheses.
     
  4. Oct 20, 2006 #3
    Aha, got it :) Okay so generally you always want to multiply out to get addition and subtraction, right?

    And I got [tex]\frac{2x^{5/2}}{5}+2x^{1/2}[/tex] which is correct :).
     
  5. Oct 20, 2006 #4

    Hootenanny

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    Yes, it is usually easier to multiply out the parentheses since you can integrate [or differentiate] each term individually. You could of course use integration by parts to find the integral directly from the factorised form but this would be far more complicated.
     
  6. Oct 20, 2006 #5
    Okay great, thanks.
     
  7. Oct 20, 2006 #6

    Hootenanny

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    My pleasure
     
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