Evaluate the following integrals: ^ 2 _0x[x]dx - Vidyadip

Question

Evaluate the following integrals:
$\int\limits^{2}_0\text{x}[\text{x}]\text{dx}$

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Answer

We have,
$\text{I}=\int\limits^{2}_0\text{x}[\text{x}]\text{dx}$
We know that,
$\text{x}[\text{x}]=\begin{cases}\text{x}\times0,&0<\text{x}<1\\\text{x}\times1,&1<\text{x}<2\end{cases}$
i.e.,
$\text{x}[\text{x}]=\begin{cases}0,&0<\text{x}<1\\\text{x},&1<\text{x}<2\end{cases}$
$\therefore\ \text{I}=\int\limits^{2}_0\text{x}[\text{x}]\text{dx}$
$=\int\limits^{1}_0\text{x}[\text{x}]\text{dx}+\int\limits^{2}_1\text{x}[\text{x}]\text{dx}$
$=\int\limits^{1}_0{0}\text{ dx}+\int\limits^{2}_1(\text{x})\text{dx}$
$=0+\Big[\frac{\text{x}^2}{2}\Big]^2_1$
$=\frac{2^2}{2}-\frac{1^2}{2}$
$=\frac{3}{2}$

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