How to show that $\int_0^x |z|\,dz = \frac12 x|x|$?

Question

Say you are integrating a simple $|z|dx$ from $0$ to $x$. How do you go about solving to get $.5 x|x|$?

Answer 1

If you can use the function $\rm{signum}(x)$ defined for $x \neq 0$ and $1$ for $x>0,$ $-1$ for $x<0,$ then (for nonzero $x$) $|x|=x \cdot \rm{signum}(x),$ and the product rule on that gives $(|x|)'=\rm{signum}(x).$With this, the product rule on $x|x|$ gives its derivative as

$$x \cdot \rm{signum}(x)+1\cdot |x|,$$ or $2|x|.$ Division by $2$ then gives $\frac{d}{dx}(.5x|x|)=|x|.$

Answer 2

$\int_0^x|z|~dz$

$=[z|z|]_0^x-\int_0^xz~d(|z|)$

$=x|x|-\int_0^xz\times\dfrac{|z|}{z}dz$

$=x|x|-\int_0^x|z|~dz$

$\therefore2\int_0^x|z|~dz=x|x|$

$\int_0^x|z|~dz=\dfrac{x|x|}{2}$