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

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To integrate ( \frac{x^2}{(x-1)(x^2+2x+1)} ) using partial fractions, first factor the denominator as ( (x-1)(x+1)^2 ). Then, express ( \frac{x^2}{(x-1)(x^2+2x+1)} ) as the sum of two fractions with unknown constants:

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

Next, clear the denominators by multiplying both sides by ( (x-1)(x+1)^2 ), then equate coefficients of like terms:

[ x^2 = A(x+1)^2 + B(x-1)(x+1) + C(x-1) ]

Solve for ( A ), ( B ), and ( C ) by substituting appropriate values of ( x ) to eliminate one of the unknowns at a time. Once you find the values of ( A ), ( B ), and ( C ), integrate each term separately to obtain the final result.

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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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