1. The problem is to find the integral of the absolute value function $|x|$ with respect to $x$. 2. Recall that the absolute value function $|x|$ is defined as: $$|x| = \begin{cases} x & \text{if } x \geq 0 \\ -x & \text{if } x < 0 \end{cases}$$ 3. To integrate $|x|$, we split the integral into two parts based on the definition: $$\int |x| \, dx = \int_{x<0} -x \, dx + \int_{x\geq0} x \, dx$$ 4. For $x < 0$: $$\int -x \, dx = -\frac{x^2}{2} + C_1$$ 5. For $x \geq 0$: $$\int x \, dx = \frac{x^2}{2} + C_2$$ 6. Combining these results, the integral of $|x|$ is: $$\int |x| \, dx = \begin{cases} -\frac{x^2}{2} + C_1 & \text{if } x < 0 \\ \frac{x^2}{2} + C_2 & \text{if } x \geq 0 \end{cases}$$ 7. Usually, we write the integral with a single constant of integration $C$ and note the piecewise nature: $$\int |x| \, dx = \frac{x|x|}{2} + C$$ This works because $x|x| = x \cdot x = x^2$ when $x \geq 0$ and $x|x| = x \cdot (-x) = -x^2$ when $x < 0$. Final answer: $$\int |x| \, dx = \frac{x|x|}{2} + C$$