# How do you integrate #(y^4+y^2-1)/(y^3+y)# using partial fractions?

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To integrate the rational function (y^4 + y^2 - 1) / (y^3 + y) using partial fractions, follow these steps:

- Factor the denominator (y^3 + y) as y(y^2 + 1).
- Express the fraction as a sum of partial fractions with undetermined coefficients:

(y^4 + y^2 - 1) / (y^3 + y) = A/y + (By + C) / (y^2 + 1)

- Multiply both sides by the denominator (y^3 + y) to clear the fractions:

y^4 + y^2 - 1 = A(y^2 + 1) + (By + C)y

- Expand and group like terms:

y^4 + y^2 - 1 = Ay^2 + A + By^2 + Cy

- Equate coefficients of like powers of y on both sides of the equation:

For y^4 term: 0 = A For y^3 term: 0 = 0 (no term on the right side) For y^2 term: 1 = A + B For y term: 0 = C

- Solve the system of equations to find the values of A, B, and C:

From equation 1: A = 0 From equation 3: B = 1 - A = 1 From equation 4: C = 0

So, the partial fraction decomposition is:

(y^4 + y^2 - 1) / (y^3 + y) = 1 / y + y / (y^2 + 1)

- Now, integrate each partial fraction separately:

∫(1 / y) dy = ln|y| ∫(y / (y^2 + 1)) dy = (1/2) ln|y^2 + 1|

- Combine the integrals:

∫(y^4 + y^2 - 1) / (y^3 + y) dy = ln|y| + (1/2) ln|y^2 + 1| + C

Where C is the constant of integration.

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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.

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