How do you integrate #int (x-5)/(x-2)^2# using partial fractions?
Partial fractions are not really necessary in this case because we can write: to divide the numerator.
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To integrate ( \frac{{x - 5}}{{(x - 2)^2}} ) using partial fractions, first express it as a sum of partial fractions:
[ \frac{{x - 5}}{{(x - 2)^2}} = \frac{A}{{x - 2}} + \frac{B}{{(x - 2)^2}} ]
Now, multiply both sides by ( (x - 2)^2 ) to clear the denominators:
[ x - 5 = A(x - 2) + B ]
Expand and equate coefficients:
[ x - 5 = Ax - 2A + B ]
From here, you can equate coefficients of like terms. Start by equating the constant terms:
[ -5 = -2A + B ]
Then equate the coefficients of ( x ):
[ 1 = A ]
Now that you have found the values of ( A ) and ( B ), substitute them back into the partial fractions:
[ \frac{{x - 5}}{{(x - 2)^2}} = \frac{1}{{x - 2}} + \frac{B}{{(x - 2)^2}} ]
[ \frac{{x - 5}}{{(x - 2)^2}} = \frac{1}{{x - 2}} + \frac{4}{{(x - 2)^2}} ]
Now, integrate each term separately:
[ \int \frac{{x - 5}}{{(x - 2)^2}} , dx = \int \frac{1}{{x - 2}} , dx + \int \frac{4}{{(x - 2)^2}} , dx ]
[ = \ln|x - 2| - \frac{4}{{x - 2}} + 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.
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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