Let [] denote the greatest integer function.

Question

Let $[\cdot]$ denote the greatest integer function. If $\displaystyle\int_0^{e^3}\left[\frac{1}{e^{x-1}}\right]dx = \alpha - \log_e2$, then $\alpha^3$ is equal to \_\_\_\_\_\_\_\_\_.

Answer: 8

Accepted Answer

Write $\dfrac{1}{e^{x-1}} = e^{1-x}$. Then $e^{1-x}\ge2$ for $x\le 1-\ln2$ (bracket $=2$); $1\le e^{1-x}<2$ for $1-\ln2<x\le1$ (bracket $=1$); $0\le e^{1-x}<1$ for $x>1$ (bracket $=0$).

$$I = \int_0^{1-\ln2}2\,dx + \int_{1-\ln2}^1 1\,dx + \int_1^{e^3}0\,dx = 2(1-\ln2) + \ln2 = 2-\ln2$$

Comparing with $\alpha-\ln2 = 2-\ln2$ gives $\alpha=2$, so $\alpha^3=8$.

Solutions · 1