How do you integrate #int (1-x^2)/((x-9)(x-2)(x-2)) # using partial fractions?
I divided the integrand into fundamental fractions.
Following denominator expansion,
Thus,
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To integrate ( \frac{1 - x^2}{(x - 9)(x - 2)^2} ) using partial fractions, follow these steps:
- Perform polynomial long division if necessary to ensure the degree of the numerator is less than the degree of the denominator.
- Factor the denominator into linear factors.
- Express the given fraction as a sum of partial fractions with undetermined coefficients.
- Solve for the coefficients by equating the original expression to the sum of the partial fractions.
- Integrate each partial fraction.
- Combine the integrals to get the final result.
Let's proceed with the partial fractions decomposition:
[ \frac{1 - x^2}{(x - 9)(x - 2)^2} = \frac{A}{x - 9} + \frac{B}{x - 2} + \frac{C}{(x - 2)^2} ]
Now, find ( A ), ( B ), and ( C ):
[ 1 - x^2 = A(x - 2)^2 + B(x - 9)(x - 2) + C(x - 9) ]
After finding the values of ( A ), ( B ), and ( C ), integrate each term separately, and then combine the integrals to get the final result.
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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.
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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