# How do you integrate #(5x^2-9x)/((x-4)(x-1)^2)# using partial fractions?

Equating coefficients we get this system of linear equations:

Adding all three equations, we find:

Adding twice the first equation to the second, we get:

Hence:

Then from the first equation:

So:

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To integrate ((5x^2-9x)/((x-4)(x-1)^2)) using partial fractions, follow these steps:

- Perform long division if necessary to ensure that the degree of the numerator is less than the degree of the denominator.
- Write the expression in the form (\frac{A}{x-4} + \frac{B}{x-1} + \frac{C}{(x-1)^2}).
- Multiply both sides by the denominator ((x-4)(x-1)^2) to clear the fractions.
- Solve for the constants (A), (B), and (C) by comparing coefficients.
- Integrate each term separately.
- Combine the results to get the final integral.

The result after integration will be:

(\frac{5}{3}\ln|x-1| - \frac{3}{x-1} - 4\ln|x-4| + 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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