Let [.] denote the greatest integer function.

Question

Let [.] denote the greatest integer function. Then the value of $\int _{0}^{3}(\frac{{e}^{x}+{e}^{-x}}{[x]!})dx$ is :

Accepted Answer

$\int _{0}^{1}\frac{{e}^{x}+{e}^{-x}}{1}\, dx+\int _{1}^{2}\frac{{e}^{x}+{e}^{-x}}{1}\, dx+\int _{2}^{3}\frac{{e}^{x}+{e}^{-x}}{2}\, dx$

= $\int _{0}^{2}{e}^{x}\, dx+\int _{0}^{2}{e}^{-x}\, dx+\frac{1}{2}\int _{2}^{3}({e}^{x}+{e}^{-x})\, dx$

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

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

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

Solutions · 1