Which antiderivative corresponds to ∫ (3x+4)/(x^2 - x) dx?

• A. 3 ln|x| - 2 ln|x-1| + C
• B. -4 ln|x| + 7 ln|x-1| + C
• C. 4 ln|x| - 7 ln|x-1| + C
• D. (1/2) ln|x| + (1/2) ln|x-1| + C

Answer: (B)

The integrand is a rational function with the denominator factors x and x−1, so use partial fractions to turn it into simple log integrals.

Write (3x+4)/(x(x−1)) as A/x + B/(x−1). Multiplying through gives 3x+4 = A(x−1) + Bx = (A+B)x − A. Matching coefficients yields A+B = 3 and −A = 4, so A = −4 and B = 7.

Now integrate term by term: ∫(−4/x) dx + ∫(7/(x−1)) dx = −4 ln|x| + 7 ln|x−1| + C.

This is valid for x ≠ 0 and x ≠ 1. The result comes directly from the unique partial-fraction decomposition, so it matches the correct antiderivative.