Question

If \[ \frac{x}{(x-1)(x^2+1)^2} = \frac{1}{4} \left[\frac{1}{x-1} - \frac{x+1}{x^2+1}\right] + y, \] then \(y =\)

• A. \(\frac{1-x}{2(x^2+1)^2}\)
• B. \(\frac{1+x}{3(x^2+1)^2}\)
• C. \(\frac{1-x}{(x^2-1)^2}\)
• D. \(\frac{1+x}{(x^2+1)^2}\)

Hint:

When decomposing rational expressions, carefully align denominators and simplify stepwise to isolate unknown parts.

Answer 1

The Correct Option is A

Step 1: Start with given expression.
\[ \frac{x}{(x-1)(x^2+1)^2} = \frac{1}{4} \left[\frac{1}{x-1} - \frac{x+1}{x^2+1}\right] + y \]

Step 2: Combine fractions on RHS.
\[ \frac{1}{4} \left[\frac{1}{x-1} - \frac{x+1}{x^2+1}\right] = \frac{(x^2+1)^2 - (x-1)(x+1)(x^2+1)}{4(x-1)(x^2+1)^2} = \dots \]

Step 3: Simplify numerator.
After simplification: \[ \frac{x}{(x-1)(x^2+1)^2} - \frac{1}{4} \left[\frac{1}{x-1} - \frac{x+1}{x^2+1}\right] = y \]

Step 4: Simplify further.
Combine terms and simplify to get: \[ y = \frac{1-x}{2(x^2+1)^2} \]

Step 5: Final conclusion.
\[ \boxed{y = \frac{1-x}{2(x^2+1)^2}} \]