How do you find #int x^3/(x^4x^21)dx# using partial fractions?
where
Therefore,
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To find ∫x^3/(x^4  x^2  1) dx using partial fractions, follow these steps:

Factor the denominator: x^4  x^2  1 = (x^2  φ)(x^2 + φ), where φ is the golden ratio (φ ≈ 1.618).

Decompose the fraction into partial fractions: x^3 / (x^4  x^2  1) = A(x + φ) + B(x  φ) + C(x^2 + φ) + D(x^2  φ)

Multiply both sides by the denominator to clear the fraction.

Equate coefficients of like terms to solve for A, B, C, and D.

Integrate each term separately.

Combine the results to get the final answer.
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To find ( \int \frac{x^3}{x^4  x^2  1} , dx ) using partial fractions, first factor the denominator. Then express the integrand as a sum of simpler fractions with undetermined coefficients.
The denominator ( x^4  x^2  1 ) factors as ( (x^2 + 1)(x^2  1) ), or ( (x^2 + 1)(x + 1)(x  1) ).
Now, we express ( \frac{x^3}{x^4  x^2  1} ) as a sum of partial fractions:
[ \frac{x^3}{x^4  x^2  1} = \frac{Ax + B}{x^2 + 1} + \frac{C}{x + 1} + \frac{D}{x  1} ]
Next, we multiply both sides by the denominator ( x^4  x^2  1 ) and simplify to solve for ( A ), ( B ), ( C ), and ( D ).
Finally, we integrate each partial fraction separately.
[ \int \frac{x^3}{x^4  x^2  1} , dx = \int \frac{Ax + B}{x^2 + 1} , dx + \int \frac{C}{x + 1} , dx + \int \frac{D}{x  1} , dx ]
[ = \int \frac{Ax}{x^2 + 1} , dx + \int \frac{B}{x^2 + 1} , dx + \int \frac{C}{x + 1} , dx + \int \frac{D}{x  1} , dx ]
[ = \frac{A}{2} \lnx^2 + 1 + B \arctan(x) + C \lnx + 1 + D \lnx  1 + C_1 ]
Where ( C_1 ) is the constant of integration.
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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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