# How do you integrate #int (2x+4)/((x+2)(x-6)(x^2+3)) dx# using partial fractions?

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To integrate ( \frac{2x+4}{(x+2)(x-6)(x^2+3)} ) using partial fractions, follow these steps:

- Factor the denominator completely: ( (x+2)(x-6)(x^2+3) ).
- Write the fraction in the form of partial fractions: ( \frac{2x+4}{(x+2)(x-6)(x^2+3)} = \frac{A}{x+2} + \frac{B}{x-6} + \frac{Cx+D}{x^2+3} ).
- Clear the denominators by multiplying both sides by the common denominator ( (x+2)(x-6)(x^2+3) ).
- After clearing denominators, equate the numerators: ( 2x+4 = A(x-6)(x^2+3) + B(x+2)(x^2+3) + (Cx+D)(x+2)(x-6) ).
- Substitute suitable values of ( x ) to solve for ( A ), ( B ), ( C ), and ( D ).
- Once you find the values of ( A ), ( B ), ( C ), and ( D ), rewrite the original integral using the partial fraction decomposition.
- Integrate each partial fraction separately.
- Finally, combine the integrals to obtain the result.

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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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