Resolve into Partial Fractions
(i) `(x^(2)+1)/((x^(2)+4)(x-2))`

Answer

To resolve the expression \(\frac{x^2 + 1}{(x^2 + 4)(x - 2)}\) into partial fractions, we will follow these steps:

### Step 1: Set Up the Partial Fraction Decomposition
We start by expressing the given fraction as a sum of simpler fractions. Since the denominator consists of a quadratic factor \((x^2 + 4)\) and a linear factor \((x - 2)\), we can write:

\[ \frac{x^2 + 1}{(x^2 + 4)(x - 2)} = \frac{Ax + B}{x^2 + 4} + \frac{C}{x - 2} \]

where \(A\), \(B\), and \(C\) are constants that we need to determine.

### Step 2: Combine the Right Side
Next, we will combine the right-hand side over a common denominator:

\[ \frac{Ax + B}{x^2 + 4} + \frac{C}{x - 2} = \frac{(Ax + B)(x - 2) + C(x^2 + 4)}{(x^2 + 4)(x - 2)} \]

### Step 3: Expand the Numerator
Now, let's expand the numerator:

\[ (Ax + B)(x - 2) + C(x^2 + 4) = Ax^2 - 2Ax + Bx - 2B + Cx^2 + 4C \]

Combining like terms gives:

\[ (A + C)x^2 + (-2A + B)x + (-2B + 4C) \]

### Step 4: Set Up the Equation
Now, we equate the numerators from both sides:

\[ x^2 + 1 = (A + C)x^2 + (-2A + B)x + (-2B + 4C) \]

### Step 5: Compare Coefficients
From the equation above, we can compare the coefficients of corresponding powers of \(x\):

1. Coefficient of \(x^2\): \(A + C = 1\) (Equation 1)
2. Coefficient of \(x\): \(-2A + B = 0\) (Equation 2)
3. Constant term: \(-2B + 4C = 1\) (Equation 3)

### Step 6: Solve the System of Equations
We now have a system of equations to solve:

1. From Equation 2, we can express \(B\) in terms of \(A\):
\[ B = 2A \]

2. Substitute \(B = 2A\) into Equation 3:
\[ -2(2A) + 4C = 1 \implies -4A + 4C = 1 \implies C = A + \frac{1}{4} \]

3. Substitute \(C\) into Equation 1:
\[ A + \left(A + \frac{1}{4}\right) = 1 \implies 2A + \frac{1}{4} = 1 \implies 2A = \frac{3}{4} \implies A = \frac{3}{8} \]

4. Now substitute \(A\) back to find \(B\) and \(C\):
\[ B = 2A = 2 \times \frac{3}{8} = \frac{3}{4} \]
\[ C = A + \frac{1}{4} = \frac{3}{8} + \frac{1}{4} = \frac{3}{8} + \frac{2}{8} = \frac{5}{8} \]

### Step 7: Write the Final Result
Now we can write the partial fraction decomposition:

\[ \frac{x^2 + 1}{(x^2 + 4)(x - 2)} = \frac{\frac{3}{8}x + \frac{3}{4}}{x^2 + 4} + \frac{\frac{5}{8}}{x - 2} \]

### Summary of the Values
Thus, the values of \(A\), \(B\), and \(C\) are:
- \(A = \frac{3}{8}\)
- \(B = \frac{3}{4}\)
- \(C = \frac{5}{8}\)