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

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

- Factor the denominator to identify its roots.
- Decompose the fraction into partial fractions based on the factors of the denominator.
- Solve for the coefficients of the partial fractions.
- Integrate each partial fraction separately.
- Combine the results to find the overall integral.

Given that the denominator (3 - 2x - x^2) factors as (-(x + 1)(x - 3)), the partial fraction decomposition is:

[\frac{{2x - 2}}{{\sqrt{3 - 2x - x^2}}} = \frac{{A}}{{\sqrt{x + 1}}} + \frac{{B}}{{\sqrt{x - 3}}}]

Now, solve for (A) and (B) by multiplying through by the common denominator:

[2x - 2 = A\sqrt{x - 3} + B\sqrt{x + 1}]

After finding (A) and (B), integrate each term separately:

[\int \frac{{A}}{{\sqrt{x + 1}}}dx = A\sqrt{x + 1} + C_1] [\int \frac{{B}}{{\sqrt{x - 3}}}dx = B\sqrt{x - 3} + C_2]

Finally, combine the results and add the constant of integration to get the overall integral:

[\int \frac{{2x - 2}}{{\sqrt{3 - 2x - x^2}}}dx = A\sqrt{x + 1} + B\sqrt{x - 3} + C]

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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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