Question Detail
Question
Integrate ∫x/(x-1) dx
Correct Answer
Option B is the correct answer.
Detailed Explanation
To integrate x/(x-1), first perform algebraic manipulation to simplify the integrand.Rewrite the numerator: x = (x-1) + 1.So, x/(x-1) = ((x-1) + 1)/(x-1) = (x-1)/(x-1) + 1/(x-1) = 1 + 1/(x-1).Now, integrate term by term:∫(1 + 1/(x-1)) dx = ∫1 dx + ∫1/(x-1) dx= x + ln|x-1| + C.Assuming x-1 > 0 (or a domain where x-1 is positive), this simplifies to x + ln(x-1) + C. This matches option B.
Hint
Simplify the integrand by adding and subtracting 1 in the numerator or by performing polynomial long division before integrating. Remember the integral of 1/u is ln|u|.