# How do you use partial fraction decomposition to decompose the fraction to integrate #x^4/((x-1)^3)#?

First perform the division.

In order to use partial fraction decomposition we must have the degree of the numerator less than the degree of the denominator.

Clear the denominators to get:

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To decompose the fraction ( \frac{x^4}{(x-1)^3} ) using partial fraction decomposition, we first factor the denominator ( (x-1)^3 ) to its linear factors. In this case, it's ( (x-1)(x-1)(x-1) ). Since the degree of the numerator ( x^4 ) is greater than the degree of the denominator ( (x-1)^3 ), we need to perform polynomial division first.

After dividing ( x^4 ) by ( (x-1)^3 ), we get a quotient and a remainder. The quotient will be a polynomial of degree less than the denominator, which is ( (x-1)^3 ). The remainder will be a polynomial of degree less than the divisor ( (x-1)^3 ).

After performing polynomial division, the expression ( \frac{x^4}{(x-1)^3} ) can be rewritten as a sum of partial fractions, each with a simpler denominator than the original. The partial fraction decomposition will look like ( \frac{A}{x-1} + \frac{B}{(x-1)^2} + \frac{C}{(x-1)^3} ).

We then find the values of constants ( A ), ( B ), and ( C ) by equating the original expression with the sum of partial fractions and solving for ( A ), ( B ), and ( C ) using algebraic methods, such as equating coefficients.

Once we have found the values of ( A ), ( B ), and ( C ), we substitute them back into the partial fraction decomposition.

Finally, we integrate each partial fraction term separately to obtain the integral of ( \frac{x^4}{(x-1)^3} ).

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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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