How do you integrate #int (x^22x1) / ((x1)^2 (x^2+1))# using partial fractions?
By converting the fraction into a sum of independent factors
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To integrate the function ( \frac{x^2  2x  1}{(x1)^2(x^2 + 1)} ) using partial fractions, follow these steps:

Factor the denominator completely. In this case, ( (x1)^2(x^2 + 1) ) is already factored.

Write the partial fraction decomposition as ( \frac{A}{x1} + \frac{B}{(x1)^2} + \frac{Cx + D}{x^2 + 1} ), where A, B, C, and D are constants to be determined.

Multiply both sides of the equation by the denominator of the original expression to clear the fractions.

After clearing the fractions, equate the numerators to find the values of A, B, C, and D.

Once you've found the values of A, B, C, and D, rewrite the original integral using the partial fraction decomposition.

Integrate each term separately.

Finally, add the integrated terms together to get the result of the original integral.
This process involves decomposing the original rational function into simpler fractions, integrating each term, and then combining them to find the integral of the original function.
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When evaluating a onesided 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 onesided 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 onesided 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 onesided 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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