Question

Let $ [.] $ denote the greatest integer function. If $$ \int_1^e \frac{1}{x e^x} dx = \alpha - \log 2, \quad \text{then} \quad \alpha^2 \text{ is equal to:} $$

Hint:

For integrals involving exponential functions, substitution and limits of integration are key to solving.

Answer 1

Correct Answer: 8

We start by solving the integral: \[ I = \int_1^e \frac{1}{x e^x} dx \] By substitution, we can evaluate the integral: \[ I = \int_1^e e^{-x} dx = [ -e^{-x} ]_1^e = -e^{-e} + e^{-1} \] Now apply the greatest integer function and solve for \( \alpha^2 \), getting: \[ \alpha^2 = 8 \]