# How do you find #int (2x)/(x^3-x^2+x-1)dx# using partial fractions?

I found:

I had to do some manipulations (in the blue frames) and get:

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To find the integral of (2x)/(x^3 - x^2 + x - 1)dx using partial fractions, follow these steps:

- Factor the denominator: x^3 - x^2 + x - 1 can be factored as (x - 1)(x^2 + 1).
- Decompose the fraction into partial fractions: (2x)/(x^3 - x^2 + x - 1) = A/(x - 1) + (Bx + C)/(x^2 + 1)
- Multiply both sides by the original denominator to clear the fractions.
- Equate coefficients of like terms.
- Solve the resulting system of equations to find the values of A, B, and C.
- Once you find the values of A, B, and C, integrate each partial fraction term separately.
- Finally, sum up the integrals of the partial fractions to find the overall integral.

The specific steps to solve the partial fractions will depend on the factors of the denominator and the coefficients involved.

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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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