Question

Sol. [Key Concept: Split by $\lbrack x!\rbrack = 1,1,2$ on $\lbrack 0,1),\lbrack 1,2),\lbrack 2,3)$ and integrate $e^{x} + e^{- x}$ on each.]

$${\int_{}^{}}_{0}^{1}\mspace{2mu}\frac{e^{x} + e^{- x}}{1}dx + {\int_{}^{}}_{1}^{2}\mspace{2mu}\frac{e^{x} + e^{- x}}{1}dx + {\int_{}^{}}_{2}^{3}\mspace{2mu}\frac{e^{x} + e^{- x}}{2}dx $$

$${\int_{}^{}}_{0}^{2}\mspace{2mu} e^{x}dx + {\int_{}^{}}_{0}^{2}\mspace{2mu} e^{- x}dx + \frac{1}{2}{\int_{}^{}}_{2}^{3}\mspace{2mu}\left( e^{x} + e^{- x} \right)dx $$

$$\left( e^{2} - 1 \right) + \frac{\left( e^{- 2} - 1 \right)}{- 1} + \frac{1}{2}\left( \left( e^{3} - e^{2} \right) - \left( e^{- 3} - e^{- 2} \right) \right) $$

$$\frac{e^{3} + e^{2}}{2} + 1 - 1 - \frac{e^{- 2}}{2} - \frac{e^{- 3}}{2} $$

$$\frac{e^{3} + e^{2} - e^{- 2} - e^{- 3}}{2} $$