How do you find #int (x^3+1)/((x^24)(x^2+1)) dx# using partial fractions?
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To find the integral of (x^3 + 1)/((x^2  4)(x^2 + 1)) dx using partial fractions, follow these steps:

Factor the denominator: (x^2  4)(x^2 + 1) = (x  2)(x + 2)(x^2 + 1).

Write the fraction as a sum of partial fractions: (x^3 + 1)/((x  2)(x + 2)(x^2 + 1)) = A/(x  2) + B/(x + 2) + (Cx + D)/(x^2 + 1).

Multiply both sides of the equation by the denominator to clear it: x^3 + 1 = A(x + 2)(x^2 + 1) + B(x  2)(x^2 + 1) + (Cx + D)(x  2)(x + 2).

Substitute appropriate values of x to solve for A, B, C, and D. You can choose values of x that make one or more terms drop out.

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

Integrate each term separately.

Combine the integrals to get the final result.
This process allows you to integrate the given expression using partial fractions.
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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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