How do you integrate #(y^2 + 1) / (y^3  1)# using partial fractions?
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To integrate (\frac{y^2 + 1}{y^3  1}) using partial fractions, follow these steps:

Factor the denominator (y^3  1): [ y^3  1 = (y  1)(y^2 + y + 1) ]

Decompose the fraction into partial fractions: [ \frac{y^2 + 1}{y^3  1} = \frac{A}{y  1} + \frac{By + C}{y^2 + y + 1} ]

Clear the fractions by multiplying through by the common denominator (y^3  1): [ y^2 + 1 = A(y^2 + y + 1) + (By + C)(y  1) ]

Expand and collect like terms: [ y^2 + 1 = Ay^2 + Ay + A + By^2  By + Cy  C ]

Group the terms with like powers of (y): [ y^2 \text{ terms}: \quad A + B = 1 ] [ y \text{ terms}: \quad A  B + C = 0 ] [ \text{constant terms}: \quad A + C = 1 ]

Solve the system of equations to find (A), (B), and (C).
From the first equation: [ A + B = 1 ] Given that (A = 1  B).
From the third equation: [ A + C = 1 ] Given that (C = 1  A).
Substituting (A = 1  B) into (A + C = 1): [ (1  B) + C = 1 ] [ C = B ]
Using this result in the second equation: [ (1  B)  B + B = 0 ] [ B = 1 ]
Then, (A = 1  B = 0) and (C = B = 1).

Rewrite the original integral using the partial fractions: [ \int \frac{y^2 + 1}{y^3  1} , dy = \int \frac{0}{y  1} , dy + \int \frac{y + 1}{y^2 + y + 1} , dy ]

Integrate each term: [ \int \frac{y + 1}{y^2 + y + 1} , dy ] can be solved by completing the square or using a trigonometric substitution.
After integrating each term, you'll get the final result for the integral.
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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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