# How do you integrate #int (x+3)/(x^2(x-1) )# using partial fractions?

without utilizing fractions that are partially expressed.

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To integrate ( \frac{x+3}{x^2(x-1)} ) using partial fractions, we first decompose the fraction into partial fractions. Then we integrate each term separately.

First, we rewrite the fraction as the sum of two fractions: [ \frac{x+3}{x^2(x-1)} = \frac{A}{x} + \frac{B}{x^2} + \frac{C}{x-1} ]

We then find the values of ( A ), ( B ), and ( C ) by multiplying both sides by the common denominator ( x^2(x-1) ) and simplifying.

Next, we integrate each term: [ \int \frac{A}{x} , dx = A \ln|x| + C_1 ] [ \int \frac{B}{x^2} , dx = -\frac{B}{x} + C_2 ] [ \int \frac{C}{x-1} , dx = C \ln|x-1| + C_3 ]

Finally, we combine the results and simplify if necessary.

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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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