How Do You Solve the Integral of (|X|)^0.5 dx?

[somebody-nobody]

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]

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.

 

[benorin]

integral of sqrt[abs[x]]

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

Proof: Consider that for x\geq 0, we have

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

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

\int_0^x \sqrt{|X|}dX =\int_0^{-t} \sqrt{|X|}dX

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

\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

putting these togeather we have

\int_0^x \sqrt{|X|}dX =\left\{\begin{array}{cc}\frac{2}{3}x\sqrt{-x}, &amp; \mbox{ if } x\leq 0\\ \frac{2}{3}x\sqrt{x},&amp;\mbox{ if }<br /> x\geq 0\end{array}\right. =\frac{2}{3}x\sqrt{|x|}​

 

[somebody-nobody]

thank you

thanks to both of you