How to demostrate that Integrate of |t|dt in {0,x} for any x in R is (1/2)x|x|
We are given to verify:
(1) $\displaystyle \int_0^x |t|\,dt=\frac{1}{2}x|x|$
The derivative form of the fundamental theorem of calculus is:
$\displaystyle \frac{d}{dx}\int_a^x f(t)\,dx=f(x)$
Use this for the left side, and on the right use the product rule and the fact that we have $\displaystyle 0\le x$.
On the right side, you have:
$\displaystyle \frac{1}{2}x|x|=\frac{1}{2}x\sqrt{x^2}$
Using the product, power and chain rules, we find:
$\displaystyle \frac{d}{dx}\left(\frac{1}{2}x\sqrt{x^2} \right)=\frac{1}{2}\left(x\frac{x}{\sqrt{x^2}}+ \sqrt{x^2} \right)=\frac{x^2}{\sqrt{x^2}}=\sqrt{x^2}=|x|$
Thus, we have shown the given result for the definite integral is valid for all real $\displaystyle x$.