Evaluate the following integrals: ^1_ -1 x|x|dx - Vidyadip

Question

Evaluate the following integrals:
$\int\limits^1_{-1}\text{x|x|}\text{dx}$

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Answer

$|\text{x}|=\begin{cases}-\text{x},&-1<\text{x}<0\\\text{x},&0<\text{x}<1\end{cases}$
$\therefore\ \text{x}|\text{x}|=\begin{cases}-\text{x}^2,&-1<\text{x}<0\\\text{x}^2,&0<\text{x}<1\end{cases}$
Now, $\int\limits^1_{-1}\text{x|x|}\text{dx}$
$=\int\limits^0_{-1}-\text{x}^2\text{ dx}+\int\limits^1_{0}\text{x}^2\text{ dx}$
$=-\int\limits^0_{-1}\text{x}^2\text{ dx}+\int\limits^1_{0}\text{x}^2\text{ dx}$
$=-\Big[\frac{\text{x}^3}{3}\Big]^0_{-1}+\Big[\frac{\text{x}^3}{3}\Big]^1_0$
$=-\Big(0+\frac{1}{3}\Big)+\Big(\frac{1}{3}-0\Big)$
$=0-\frac{1}{3}+\frac{1}{3}-0$
$=0$

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