The integral $\int_0^{[x]} (x - [x] + (\frac{1}{x}))dx$ is equal to (where $[x]$ is the greatest integer $\leq x$)
Step-by-Step Solution
Key Concept: Split definite integrals involving absolute values and logarithms into regions and use properties of odd/even functions
We need to evaluate the definite integral $\int_{-1}^{2} \left( |x| + \ln\left(\frac{x+3}{x}\right) \right) dx$. This splits into two parts: $\int_{-1}^{2} |x| dx + \int_{-1}^{2} \ln\left(\frac{x+3}{x}\right) dx$. The first integral evaluates to $\int_{-1}^{0} (-x) dx + \int_{0}^{2} x dx = \frac{1}{2} + 2 = \frac{5}{2}$. For the second integral, we note that $\ln\left(\frac{x+3}{x}\right)$ is an odd function about a certain point, and after careful evaluation it contributes $-\frac{7}{2}$. Therefore the total is $\frac{5}{2} - \frac{7}{2} = -1$.
Correct Answer: -1