Question Detail

First, simplify the integrand: $\frac{2x+1}{x} = \frac{2x}{x} + \frac{1}{x} = 2 + \frac{1}{x}$.Now, integrate from $1$ to $3a$:$\int_1^{3a} \left(2 + \frac{1}{x}\right) dx = [2x + \ln|x|]_1^{3a}$$= (2(3a) + \ln(3a)) - (2(1) + \ln(1))$$= 6a + \ln(3a) - 2 - 0$$= 6a + \ln 3 + \ln a - 2$.Comparing this result with the given options, there appears to be a mismatch. The question's intended answer is marked as option B, $4a + \ln 3$. However, based on the calculation, $6a + \ln 3 + \ln a - 2$ does not simplify to $4a + \ln 3$. This suggests a potential discrepancy in the question or options provided in the source material.