Question Detail
Question
Given that a is a positive constant, evaluate ∫(from a to 3a) (2x+1)/x dx
Correct Answer
Option B is the correct answer.
Detailed Explanation
First, simplify the integrand: (2x+1)/x = 2x/x + 1/x = 2 + 1/x.Now, integrate the simplified expression:∫(2 + 1/x) dx = 2x + ln|x|.Next, evaluate the definite integral from a to 3a:[2(3a) + ln|3a|] - [2(a) + ln|a|]Since 'a' is a positive constant, |a|=a and |3a|=3a:= (6a + ln(3a)) - (2a + ln(a))= 6a - 2a + ln(3a) - ln(a)= 4a + ln(3a/a)= 4a + ln(3).This matches option B.
Hint
Split the integrand into separate terms before integrating. Remember the properties of logarithms for simplification after evaluating the definite integral.