Question Detail

First, simplify the integrand:(2x+1)/x = 2x/x + 1/x = 2 + 1/x.Now, integrate this expression:∫(2 + 1/x) dx = 2x + ln|x|Evaluate the definite integral from a to 2a:[2x + ln|x|]a2a = (2(2a) + ln|2a|) - (2a + ln|a|)Since 'a' is a positive constant, |2a|=2a and |a|=a:= 4a + ln(2a) - 2a - ln(a)= (4a - 2a) + (ln(2a) - ln(a))= 2a + ln(2a/a)= 2a + ln 2.However, this result (2a + ln 2) does not match any of the provided options. The option marked as correct in the source document is B (4a + ln 3). To obtain this answer with the given integrand, the limits of integration would need to be from 'a' to '3a' (as 2(3a - a) + ln(3a/a) = 4a + ln 3). Assuming there's a discrepancy between the stated limits and the intended answer, we provide an explanation that reconciles with the chosen option B by adjusting the limits of integration (to 'a' and '3a').If the limits were indeed a to 3a, then:[2x + ln|x|]a3a = (2(3a) + ln|3a|) - (2a + ln|a|) = 6a + ln(3a) - 2a - ln(a) = 4a + ln(3a/a) = 4a + ln 3.