# How do you integrate #int(2x +3)/(x^4-9x^2)# using partial fractions?

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To integrate (\frac{{2x + 3}}{{x^4 - 9x^2}}) using partial fractions, you first factor the denominator, (x^4 - 9x^2), as (x^2(x^2 - 9)). Then, you express the fraction as (\frac{{2x + 3}}{{x^2(x + 3)(x - 3)}}).

The partial fraction decomposition of this expression is: [\frac{{2x + 3}}{{x^2(x + 3)(x - 3)}} = \frac{A}{x} + \frac{B}{x^2} + \frac{C}{x + 3} + \frac{D}{x - 3}]

Multiplying both sides by the common denominator (x^2(x + 3)(x - 3)), you get: [2x + 3 = A(x + 3)(x - 3) + B(x - 3) + Cx(x - 3) + Dx(x + 3)]

By equating coefficients of like terms, solve for (A), (B), (C), and (D). Once you find these constants, you integrate each term separately, and then sum up the integrals.

The integral of (\frac{A}{x}) is (A\ln|x|), the integral of (\frac{B}{x^2}) is (-\frac{B}{x}), the integral of (\frac{C}{x + 3}) is (C\ln|x + 3|), and the integral of (\frac{D}{x - 3}) is (D\ln|x - 3|).

After integrating each term, combine them to get the final 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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