# Math Help - [SOLVED] Integral of abs(x)^(1/2)

1. ## [SOLVED] Integral of abs(x)^(1/2)

Hello there,

I was wondering how to integrate $|x|^{1/2}$. I don't think that $\frac{2|x|^{3/2}}{3} + C$ is correct.

Thank you!

2. Hello there,

Since:

$|x|^{1/2} = \left\{\begin{array}{cc}\sqrt{x},&\mbox{ if }
x\geq 0\\\sqrt{-x}, & \mbox{ if } x < 0\end{array}\right.$

Therefore:

$\\\int |x|^{1/2} = \left\{\begin{array}{cc}\int x^{\frac{1}{2}} \ dx,&\mbox{ if }
x\geq 0\\\int (-x)^{\frac{1}{2}} \ dx, & \mbox{ if } x < 0\end{array}\right.$

For $x < 0$, let $u = -x \Rightarrow -du = dx$ and take the antiderivative.

$\\\int |x|^{1/2} = \left\{\begin{array}{cc}\frac{2x^{3/2}}{3} + C,&\mbox{ if }
x\geq 0\\-\frac{2(-x)^{3/2}}{3} + C, & \mbox{ if } x < 0\end{array}\right.$