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:
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Factor the denominator (y^3 - 1): [ y^3 - 1 = (y - 1)(y^2 + y + 1) ]
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Decompose the fraction into partial fractions: [ \frac{y^2 + 1}{y^3 - 1} = \frac{A}{y - 1} + \frac{By + C}{y^2 + y + 1} ]
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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) ]
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Expand and collect like terms: [ y^2 + 1 = Ay^2 + Ay + A + By^2 - By + Cy - C ]
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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 ]
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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).
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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 ]
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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 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.
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.
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.
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.
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