# How do you integrate #int (4x-1)/( x^2(x-4))# using partial fractions?

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To integrate ( \frac{4x - 1}{x^2(x - 4)} ) using partial fractions, first factor the denominator. Then express the fraction as the sum of simpler fractions.

( \frac{4x - 1}{x^2(x - 4)} = \frac{A}{x} + \frac{B}{x^2} + \frac{C}{x - 4} )

Next, clear the denominators by multiplying both sides by ( x^2(x - 4) ).

( 4x - 1 = A(x)(x - 4) + B(x - 4) + C(x^2) )

Now, solve for ( A ), ( B ), and ( C ) by equating coefficients of corresponding terms.

Once you have found the values of ( A ), ( B ), and ( C ), you can rewrite the original expression as:

( \frac{4x - 1}{x^2(x - 4)} = \frac{A}{x} + \frac{B}{x^2} + \frac{C}{x - 4} )

Then integrate each term separately.

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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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