How do you integrate #int (x^2+1)/((x3)(x^3x^2)) dx# using partial fractions?
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To integrate ( \frac{{x^2 + 1}}{{(x  3)(x^3  x^2)}} ) using partial fractions, follow these steps:

Factor the denominator completely. ( (x  3)(x^3  x^2) = (x  3)x^2(x  1) )

Write the partial fraction decomposition. ( \frac{{x^2 + 1}}{{(x  3)(x^3  x^2)}} = \frac{A}{x  3} + \frac{B}{x} + \frac{C}{x^2} + \frac{D}{x  1} )

Clear the fractions by multiplying both sides by the denominator. ( x^2 + 1 = A(x)(x^2)(x  1) + B(x  3)(x^2)(x  1) + C(x  3)(x  1) + D(x  3)(x)(x  1) )

Solve for the constants ( A ), ( B ), ( C ), and ( D ) by equating coefficients of like terms.

After finding the values of ( A ), ( B ), ( C ), and ( D ), rewrite the original integral using the partial fractions.

Integrate each term separately.

Sum up the integrated terms.
The integration of ( \frac{{x^2 + 1}}{{(x  3)(x^3  x^2)}} ) using partial fractions involves these steps to find the solution.
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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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