# How do you integrate #int [1/ (s (s - 1)^2)]# using partial fractions?

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To integrate ( \int \frac{1}{s(s - 1)^2} ) using partial fractions, we first express the given fraction as a sum of simpler fractions.

Let's denote the given fraction as ( \frac{1}{s(s - 1)^2} ). We aim to express it in the form:

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

where ( A ), ( B ), and ( C ) are constants we need to determine.

To find ( A ), ( B ), and ( C ), we clear the denominators by multiplying both sides of the equation by ( s(s - 1)^2 ):

[ 1 = A(s - 1)^2 + Bs(s - 1) + Cs ]

Expanding and simplifying:

[ 1 = As^2 - 2As + A + Bs^2 - Bs + Cs ]

Now, we equate the coefficients of corresponding terms on both sides of the equation:

For ( s^2 ): ( A + B = 0 ) For ( s ): ( -2A - B + C = 0 ) For the constant term: ( A = 1 )

Solving these equations simultaneously will give us the values of ( A ), ( B ), and ( C ).

From the first equation, we get ( A = -B ). Plugging this into the second equation:

[ -2(-B) - B + C = 0 ] [ 2B - B + C = 0 ] [ B + C = 0 ] [ C = -B ]

Now, substituting ( A = 1 ) and ( C = -B ) into ( A = -B ), we find ( A = -1 ) and ( B = 1 ).

So, the partial fraction decomposition is:

[ \frac{1}{s(s - 1)^2} = \frac{-1}{s} + \frac{1}{s - 1} - \frac{1}{(s - 1)^2} ]

Now, we can integrate each term separately.

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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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