Integral (|x|)^0.5dx

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
 

radou

Homework Helper
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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
I suggest you look at what the functions |x|^(1/2) and x^(1/2) look like.
 
Last edited:

benorin

Homework Helper
1,057
7
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:
thank you

thanks to both of you
 

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