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Integral (|x|)^0.5dx

  1. Dec 12, 2006 #1
    I need help solving integral

    (|X|)^0.5dx

    is it sam integral as (x)^0.5 except that i will need to integrate from 0 to x since it is absolute value of x

    Thanks
     
  2. jcsd
  3. Dec 12, 2006 #2

    radou

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    Homework Helper

    I suggest you look at what the functions |x|^(1/2) and x^(1/2) look like.
     
    Last edited: Dec 12, 2006
  4. Dec 12, 2006 #3

    benorin

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    Homework Helper

    integral of sqrt[abs[x]]

    Answer: [tex]\int_0^x \sqrt{|X|}dX = \frac{2}{3}x\sqrt{|x|}[/tex]

    Proof: Consider that for [tex]x\geq 0,[/tex] we have

    [tex]\int_0^x \sqrt{|X|}dX =\int_0^x \sqrt{X}dX = \frac{2}{3}x\sqrt{x},\mbox{ for }x\geq 0[/tex].

    Also, if [tex]x\leq 0,[/tex], set [tex]t=-x[/tex] so that [tex]t\geq 0,[/tex] and we have

    [tex]\int_0^x \sqrt{|X|}dX =\int_0^{-t} \sqrt{|X|}dX[/tex]

    now let [tex]u=-X[/tex] so that [tex]du=-dX[/tex] and [tex]0\leq X\leq -t[/tex] becomes [tex]0\leq u\leq t[/tex] and the integral becomes

    [tex]\int_0^{-t} \sqrt{|X|}dX = -\int_0^{t} \sqrt{|-u|}du = -\int_0^{t} \sqrt{u}du = -\frac{2}{3}t\sqrt{t}= \frac{2}{3}x\sqrt{-x},\mbox{ for }x\leq 0[/tex]

    putting these togeather we have

    [tex]\int_0^x \sqrt{|X|}dX =\left\{\begin{array}{cc}\frac{2}{3}x\sqrt{-x}, & \mbox{ if } x\leq 0\\ \frac{2}{3}x\sqrt{x},&\mbox{ if }
    x\geq 0\end{array}\right. =\frac{2}{3}x\sqrt{|x|}[/tex]​
     
    Last edited: Dec 12, 2006
  5. Dec 12, 2006 #4
    thank you

    thanks to both of you
     
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