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

Answer 1

#31/49*Ln(x-2)-80/49*ln(x-9)-3/7*(x-2)^(-1)+C#

I divided the integrand into fundamental fractions.

#(1-x^2)/[(x-9)*(x-2)^2]=A/(x-9)+B/(x-2)+C/(x-2)^2#

Following denominator expansion,

#A*(x-2)^2+B*(x-2)(x-9)+C*(x-9)=1-x^2#
After setting #x=2#, #-7C=-3#, so #C=3/7#
After setting #x=9#, #49A=-80#, so #A=-80/49#
After setting #x=1#, #A+8B-8C=0#, so #B=1/8*(8C-A)=31/49#

Thus,

#int (1-x^2)/[(x-9)*(x-2)^2]*dx#
=#-80/49*int (dx)/(x-9)+31/49*int (dx)/(x-2)+3/7*int (dx)/(x-2)^2#
=#31/49*Ln(x-2)-80/49*ln(x-9)-3/7*(x-2)^(-1)+C#
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Answer 2

To integrate ( \frac{1 - x^2}{(x - 9)(x - 2)^2} ) using partial fractions, follow these steps:

  1. Perform polynomial long division if necessary to ensure the degree of the numerator is less than the degree of the denominator.
  2. Factor the denominator into linear factors.
  3. Express the given fraction as a sum of partial fractions with undetermined coefficients.
  4. Solve for the coefficients by equating the original expression to the sum of the partial fractions.
  5. Integrate each partial fraction.
  6. 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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Answer from HIX Tutor

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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