Question

What is the integration of $\left|x\right|$?

Answer 1

Find the integration of $\left|x\right|$.

$\begin{array}{rcl}\left|x\right|& =& \left\{\begin{array}{l}x,x\ge 0\\ -x,x<0\end{array}\right.\\ & \Rightarrow & \int \left|x\right|dx=\left\{\begin{array}{l}\int x,x\ge 0\\ -\int x,x<0\end{array}\right.\\ & \Rightarrow & \int \left|x\right|dx=\left\{\begin{array}{c}\frac{x^2}{2}+C,x\ge 0\\ \frac{-x^2}{2}+C,x<0\end{array}\right.\end{array}\left\{\because \int x^{n}dx=\frac{x^{n+1}}{n+1}+C\right\}$

Hence, the integration of is $\int \left|x\right|dx=\left\{\begin{array}{l}\frac{x^2}{2}+C,x\ge 0\\ \frac{-x^2}{2}+C,x<0\end{array}\right.$