Question

If \( \int \frac{dx}{(x-1)^2(x-3)^2 = \sqrt{f(x)} + c \), then \( f(-1) - f(0) = \)}

• A. -3
• B. -4
• C. -2
• D. -1

Hint:

For integrals involving repeated quadratic factors, use partial fractions and evaluate with limits.

Answer 1

The Correct Option is D

Let \( f(x) = \left( \int \frac{1}{(x-1)^2(x-3)^2} dx \right)^2 \). Let’s use partial fractions: \[ \frac{1}{(x-1)^2(x-3)^2} = \frac{A}{x-1} + \frac{B}{(x-1)^2} + \frac{C}{x-3} + \frac{D}{(x-3)^2} \] Compute the integral, substitute values at \( x = -1 \) and \( x = 0 \), evaluate \( f(-1) - f(0) = -1 \).