How do you integrate #int (4x^3) / (x^3 + 2x^2  x  2)# using partial fractions?
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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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When evaluating a onesided 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.
When evaluating a onesided 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.
When evaluating a onesided 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.
When evaluating a onesided 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.
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