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 of two simpler fractions. Use a system of equations.
$\frac{4 x^{2} - 21 x - 40}{x^{3} - x^{2} - 20 x}$

$\frac{2}{x} - \frac{1}{x - 5} - \frac{3}{x + 4}$

$\frac{1}{x} - \frac{2}{x^{2} - x - 20}$

$\frac{2}{x} + \frac{1}{x + 5} - \frac{3}{x - 4}$

$\frac{2}{x} - \frac{1}{x - 5} + \frac{3}{x + 4}$

Verified step by step guidance

Factorize the denominator of the given rational function $ x^3 - x^2 - 20x $. Start by factoring out $ x $, which gives $ x(x^2 - x - 20) $. Then, factorize $ x^2 - x - 20 $ into $ (x - 5)(x + 4) $. The denominator becomes $ x(x - 5)(x + 4) $.

Rewrite the given rational function $ \frac{4x^2 - 21x - 40}{x^3 - x^2 - 20x} $ as $ \frac{4x^2 - 21x - 40}{x(x - 5)(x + 4)} $. The goal is to express this as a sum of simpler fractions: $ \frac{A}{x} + \frac{B}{x - 5} + \frac{C}{x + 4} $.

Set up the equation $ \frac{4x^2 - 21x - 40}{x(x - 5)(x + 4)} = \frac{A}{x} + \frac{B}{x - 5} + \frac{C}{x + 4} $. Multiply through by the common denominator $ x(x - 5)(x + 4) $ to eliminate the denominators, resulting in $ 4x^2 - 21x - 40 = A(x - 5)(x + 4) + Bx(x + 4) + Cx(x - 5) $.

Expand each term on the right-hand side: $ A(x - 5)(x + 4) = A(x^2 - x - 20) $, $ Bx(x + 4) = B(x^2 + 4x) $, and $ Cx(x - 5) = C(x^2 - 5x) $. Combine all terms to get $ 4x^2 - 21x - 40 = A(x^2 - x - 20) + B(x^2 + 4x) + C(x^2 - 5x) $.

Group like terms ($ x^2 $, $ x $, and constant terms) on the right-hand side and equate coefficients with the left-hand side. Solve the resulting system of equations for $ A $, $ B $, and $ C $. Substitute these values back into $ \frac{A}{x} + \frac{B}{x - 5} + \frac{C}{x + 4} $ to express the original function as a sum of simpler fractions.