If `[dot]a n d\ {dot}` denote respectively the greatest integer and fractional part functional respectively, evaluate the following integral: `int_0^1{x}dx`

Explanation

We have,
`
\begin{array}{l}
\int_{0}^{1}\{x\} d x=\int_{0}^{1}(x-[x]) d x=\int_{0}^{1} x d x- \\
\int_{0}^{1}[x] d x \\
=\left(\frac{x^{2}}{2}\right)_{0}^{1}-\left(\int_{0}^{1}[x] d x\right) \\
=\frac{1}{2}(1-0)-\left(\int_{0}^{1} 0 d x\right) \\
=0.5
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