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

#int (x^2+x)/((x+2)(x-1)^2) dx =2/9 ln abs(x+2) + 7/9 ln abs(x-1) - 2/(3(x-1)) + C#

We know that we're searching for a partial fraction decomposition of the following form because the denominator has already been factored for us:

We obtain the following system of linear equations by equating coefficients:

When we combine the three equations, we obtain:

After deducting the second equation from the initial one, we obtain:

So:

Hence:

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To integrate (\frac{x^2 + x}{(x+2)(x-1)^2}) using partial fractions, first express the fraction as the sum of partial fractions. You would typically start by writing it in the form:

(\frac{x^2 + x}{(x+2)(x-1)^2} = \frac{A}{x+2} + \frac{B}{x-1} + \frac{C}{(x-1)^2})

Then, find the values of (A), (B), and (C) by equating the numerators:

(x^2 + x = A(x-1)^2 + B(x+2)(x-1) + C(x+2))

After finding the values of (A), (B), and (C), integrate each term separately, as they become simpler to integrate once expressed as partial fractions.

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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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