Please explain the integration of greatest integer function in detail by taking examples of different powers of x. Don't give any link.

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Please find below the solution to the asked query:

$$\begin{gathered} \mathrm{Suppose} \mathrm{we} \mathrm{have} \\ \mathrm{I}={\int }_1^4\left[\mathrm{x}\right].\mathrm{dx} \\ \mathrm{Now} \mathrm{definition} \mathrm{of} \left[\mathrm{x}\right] \mathrm{changes} \mathrm{at} \mathrm{integers}, \mathrm{hence} \\ \mathrm{I}={\int }_0^1\left[\mathrm{x}\right].\mathrm{dx}+{\int }_1^2\left[\mathrm{x}\right].\mathrm{dx}+{\int }_2^3\left[\mathrm{x}\right].\mathrm{dx}+{\int }_3^4\left[\mathrm{x}\right].\mathrm{dx} \\ ={\int }_0^{1}0.\mathrm{dx}+{\int }_1^{2}1.\mathrm{dx}+{\int }_2^{3}2.\mathrm{dx}+{\int }_3^{4}3.\mathrm{dx} \\ =0+{\left[\mathrm{x}\right]}_1^2+2{\left[\mathrm{x}\right]}_2^3+3{\left[\mathrm{x}\right]}_3^4 \\ =\left[2-1\right]+2\left[3-2\right]+3\left[4-3\right] \\ =1+2+3 \\ =6 \left[\mathrm{Answer}\right] \end{gathered}$$

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