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How do you integrate #int (4x^3) / (x^3 + 2x^2 - x - 2)# using partial fractions?

Answer 1

Elijah Abram

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Answer 2

Charles Anctil

To integrate ∫(4x^3) / (x^3 + 2x^2 - x - 2) using partial fractions:

Factor the denominator: x^3 + 2x^2 - x - 2 = (x + 1)(x^2 + x - 2) = (x + 1)(x + 2)(x - 1).
Write the partial fraction decomposition: (4x^3) / (x^3 + 2x^2 - x - 2) = A / (x + 1) + B / (x + 2) + C / (x - 1).
Multiply both sides by the denominator: 4x^3 = A(x + 2)(x - 1) + B(x + 1)(x - 1) + C(x + 1)(x + 2).
Expand and equate coefficients: For x = -2: A = -2. For x = -1: B = 4. For x = 1: C = -2.
Substitute the values of A, B, and C back into the partial fraction decomposition.
Integrate each term separately.
Sum up the integrals to get the final result.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.