# How do you use partial fraction decomposition to decompose the fraction to integrate #2/(x^3-x^2)#?

Factor the denominator:

So

And we find:

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To decompose the fraction ( \frac{2}{x^3 - x^2} ) using partial fraction decomposition, follow these steps:

- Factorize the denominator: ( x^3 - x^2 = x^2(x - 1) ).
- Write the fraction in the form ( \frac{A}{x} + \frac{B}{x^2} + \frac{C}{x - 1} ).
- Multiply both sides by the denominator ( x^3 - x^2 ) to get rid of the fraction.
- Solve for ( A ), ( B ), and ( C ) by comparing the coefficients of like terms.
- Integrate each term separately.

The partial fraction decomposition of ( \frac{2}{x^3 - x^2} ) is ( \frac{2}{x^3 - x^2} = \frac{2}{x} - \frac{2}{x^2} - \frac{2}{x - 1} ).

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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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