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

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To integrate (\frac{{4x - 2}}{{3(x - 1)^2}}) using partial fractions, you first express the rational function as the sum of two or more simpler fractions.

- Factor the denominator: (3(x - 1)^2) factors into (3(x - 1)(x - 1)).
- Write the partial fraction decomposition: (\frac{{4x - 2}}{{3(x - 1)^2}} = \frac{{A}}{{3(x - 1)}} + \frac{{B}}{{3(x - 1)^2}}).
- Clear the denominators by multiplying both sides of the equation by (3(x - 1)^2).
- Solve for (A) and (B) by comparing coefficients.
- Integrate each partial fraction separately.
- Combine the results to find the final integrated expression.

After solving for (A) and (B), you should get:

[A = 2 \quad \text{and} \quad B = \frac{2}{3}]

The integral of (\frac{{4x - 2}}{{3(x - 1)^2}}) using partial fractions becomes:

[\int{\frac{{4x - 2}}{{3(x - 1)^2}}dx} = \int{\frac{{2}}{{3(x - 1)}}dx} + \int{\frac{{\frac{2}{3}}}{{3(x - 1)^2}}dx}]

[= \frac{2}{3}\ln{|x - 1|} - \frac{2}{9(x - 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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