12. Techniques of Integration

Partial Fractions

12. Techniques of Integration

Partial Fractions

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Multiple Choice

Give the partial fraction decomposition for the following expression using strategic substitutions for $x$ .
$\frac{4 x^{2} - 8 x - 8}{x (x - 2) (x + 2)}$

$\frac{2}{x} - \frac{1}{x - 2} - \frac{3}{x + 2}$

$\frac{2}{x} - \frac{1}{x - 2} + \frac{3}{x + 2}$

$\frac{4}{x^{}} - \frac{8}{x - 2} - \frac{8}{x + 2}$

$\frac{4}{x^{}} - \frac{8}{x - 2} + \frac{8}{x + 2}$

Verified step by step guidance

Step 1: Recognize that the given expression is a rational function, and the goal is to decompose it into partial fractions. The denominator is already factored as $ x(x-2)(x+2) $. This is essential for setting up the partial fraction decomposition.

Step 2: Write the general form of the partial fraction decomposition. For $ \frac{4x^2 - 8x - 8}{x(x-2)(x+2)} $, assume $ \frac{4x^2 - 8x - 8}{x(x-2)(x+2)} = \frac{A}{x} + \frac{B}{x-2} + \frac{C}{x+2} $, where $ A, B, $ and $ C $ are constants to be determined.

Step 3: Multiply through by the denominator $ x(x-2)(x+2) $ to eliminate the fractions. This gives $ 4x^2 - 8x - 8 = A(x-2)(x+2) + Bx(x+2) + Cx(x-2) $. Expand each term on the right-hand side.

Step 4: Combine like terms on the right-hand side to match the polynomial on the left-hand side. This will result in a system of equations for $ A, B, $ and $ C $ by equating the coefficients of $ x^2 $, $ x $, and the constant terms.

Step 5: Solve the system of equations to find the values of $ A, B, $ and $ C $. Substitute these values back into the partial fraction decomposition $ \frac{A}{x} + \frac{B}{x-2} + \frac{C}{x+2} $ to complete the decomposition.