Question

If \[ \int \frac{3x + 1}{(x-3)(x-5)} \, dx = \int \frac{-5}{(x-3)} \, dx + \int \frac{B}{(x-5)} \, dx, \] then the value of \( B \) is:

• A. 3
• B. 4
• C. 6
• D. 8

Hint:

When decomposing rational functions into partial fractions, equate the numerators of both sides after multiplying through by the denominator.

Answer 1

The Correct Option is B

Step 1: Break the integrand into partial fractions, where we need to find the value of \( B \).
Step 2: Use the method of partial fractions to decompose \( \frac{3x + 1}{(x-3)(x-5)} \) into two terms: \[ \frac{3x + 1}{(x-3)(x-5)} = \frac{-5}{(x-3)} + \frac{B}{(x-5)}. \] Step 3: By comparing the coefficients, we find that \( B = 4 \).

Final Answer: \[ \boxed{4} \]