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

The answer is

Reminder

Now let's break it down into partial fractions.

Compare the numerators; the denominators are the same.

Consequently,

So,

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To integrate ( \frac{1}{x(x^2 - 1)^2} ) using partial fractions, first factor the denominator completely: ( x(x^2 - 1)^2 = x(x - 1)^2(x + 1)^2 ).

Then, express the fraction as a sum of partial fractions:

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

Now, find the values of ( A ), ( B ), ( C ), ( D ), and ( E ) by finding a common denominator and equating coefficients. Then, integrate each partial fraction separately.

After integrating, the final answer will be in terms of ( \ln |x| ), ( \ln |x - 1| ), ( \frac{1}{x - 1} ), ( \ln |x + 1| ), and ( \frac{1}{x + 1} ), along with constants 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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