Find an integral formula involving the absolute value function

by RoRi

October 15, 2015

Prove that

$\int_0^x (t + |t|)^2 \, dt = \frac{2x^2}{3} (x + |x|)$

for all $x \in \mathbb{R}$ .

Proof. We consider two cases:
Case #1: If $x \geq 0$ then $t = |t|$ for all $t \in [0,x]$ so,

$\begin{align*} \int_0^x (t + |t|)^2 \, dt &= \int_0^x 4t^2 \, dt\\ &= \frac{4}{3} x^3 \\ &= \frac{2}{3} x^2 (2x) \\ &= \frac{2}{3} x^2 (x + |x|). \end{align*}$

The final equality follows since $x = |x|$ for $x \geq 0$ .
Case #2: If $x < 0$ . Then $|t| = -t$ for all $t \in [x,0]$ . So,

$\begin{align*} \int_0^x (t + |t|)^2 \, dt &= \int_0^x (t - t)^2 \, dt \\ &= 0 \\ &= \frac{2}{3} x^2 (x + |x|) \end{align*}$

since $x + |x| = 0$ for all $x < 0. \qquad \blacksquare$

Stumbling Robot

Find an integral formula involving the absolute value function

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