# How do you integrate #(x+2)/(2x^3-8x)# using partial fractions?

# int \ (x+2)/(2x^3-8x) \ dx = 1/4ln|(x-2)/x| + C#

We seek:

We can write as:

We can now decompose the integrand in to partial fractions:

Leading to the identity:

So we can now write:

Which now consists of standard integral so we integrate to get:

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To integrate the expression (x+2)/(2x^3 - 8x) using partial fractions, follow these steps:

- Factor the denominator: 2x^3 - 8x = 2x(x^2 - 4) = 2x(x - 2)(x + 2).
- Express the fraction as a sum of partial fractions: (x+2)/(2x^3-8x) = A/x + B/(x-2) + C/(x+2).
- Multiply both sides of the equation by the denominator to clear the fractions: (x + 2) = A(x - 2)(x + 2) + Bx(x + 2) + Cx(x - 2).
- Solve for A, B, and C by equating coefficients of like terms.
- Once you find the values of A, B, and C, rewrite the original fraction with these values.
- Now, integrate each term separately.
- Finally, combine the integrals and simplify the result.

The specific values of A, B, and C can be determined by equating coefficients or by using other appropriate methods.

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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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