12. Techniques of Integration

Partial Fractions

12. Techniques of Integration

Partial Fractions

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Multiple Choice

Express the rational function as a sum or difference or simpler rational expressions.
$\frac{4}{x^{3} - 3 x^{2} - x + 3}$

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

$\frac{4}{x^{3}} - \frac{4}{3 x^{2}} - \frac{4}{x + 3}$

$\frac{4}{x^{3}} - \frac{4}{3 x^{2}} + \frac{4}{x + 3}$

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

Verified step by step guidance

Factorize the denominator of the given rational function $ \frac{4}{x^3 - 3x^2 - x + 3} $. Use grouping to rewrite the denominator as $ (x^2(x - 3)) - (1(x - 3)) $, which simplifies to $ (x^2 - 1)(x - 3) $. Further factorize $ x^2 - 1 $ as $ (x - 1)(x + 1) $. The fully factorized denominator is $ (x - 3)(x - 1)(x + 1) $.

Rewrite the rational function as $ \frac{4}{(x - 3)(x - 1)(x + 1)} $. To express this as a sum of simpler fractions, use partial fraction decomposition: $ \frac{4}{(x - 3)(x - 1)(x + 1)} = \frac{A}{x - 3} + \frac{B}{x - 1} + \frac{C}{x + 1} $, where $ A $, $ B $, and $ C $ are constants to be determined.

Multiply through by the denominator $ (x - 3)(x - 1)(x + 1) $ to eliminate the fractions: $ 4 = A(x - 1)(x + 1) + B(x - 3)(x + 1) + C(x - 3)(x - 1) $. Expand each term on the right-hand side.

Group like terms and equate coefficients of $ x^2 $, $ x $, and the constant term on both sides of the equation. Solve the resulting system of equations to find the values of $ A $, $ B $, and $ C $.

Substitute the values of $ A $, $ B $, and $ C $ back into the partial fraction decomposition to express the original rational function as a sum of simpler rational expressions. Simplify each term if necessary.