Question

$\int {\left|x\right|}^{3}dx$ is equal to

(a) $\frac{-x^4}{4}+C$

(b) $\frac{{\left|x\right|}^4}{4}+C$

(c) $\frac{x^4}{4}+C$

(d) none of these

Answer 1

(d) none of these

$$\begin{gathered} \int {\left|x\right|}^{3}dx \\ \left|x\right|=\left\{\begin{array}{l}x,x\ge 0\\ -x,x<0\end{array}\right. \\ \mathrm{Case}1: \\ \mathrm{When}x\ge 0 \\ \therefore \int {\left|x\right|}^{3}dx \\ =\int x^{3}dx \\ =\frac{x^4}{4}+C \\ \mathrm{Case}2: \\ x<0 \\ \int {\left|x\right|}^{3}dx \\ =-\int x^{3}dx \\ =\frac{-x^4}{4}+C \end{gathered}$$