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

Making partial fractions as follows

Comparing corresponding coefficients on both sides we get

Now,

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To integrate (\frac{x}{x^3-x^2-2x+2}) using partial fractions, follow these steps:

- Factor the denominator: (x^3-x^2-2x+2 = (x-1)(x^2-2)).
- Express the fraction as the sum of two partial fractions: (\frac{x}{(x-1)(x^2-2)} = \frac{A}{x-1} + \frac{Bx + C}{x^2-2}).
- Clear the denominators by multiplying both sides by ((x-1)(x^2-2)).
- Simplify and solve for (A), (B), and (C) by comparing coefficients.
- Once you have the values of (A), (B), and (C), rewrite the fraction in terms of the partial fractions.
- Integrate each partial fraction separately.
- Combine the results to obtain the final integrated expression.

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