Mathematics - Definite Integration Question with Solution | TestHub

$I={\int }_0^{10}\left[x\right]·e^{\left[x\right]-x+1}$

$I={\int }_0^{1}0dx+{\int }_1^{2}1·e^{2-x}+{\int }_2^{3}2·e^{3-x}+\dots ..+{\int }_9^{10}9·e^{10-x}dx$

$\Rightarrow I=\sum _{n=0}^9{\int }_n^{n+1}n·e^{n+1-x}dx$

$=-\sum _{n=0}^{9}n{\left(e^{n+1-x}\right)}_n^{n+1}$

$=-\sum _{n=0}^{9}n·\left(e^0-e^1\right)$

$=\left(e-1\right)\sum _{n=0}^{9}n$

$=\left(e-1\right)·\frac{9·10}{2}$

$=45\left(e-1\right)$