# Evaluate the integral # int \ (4x^3-7x)/(x^4-5x^2+4) \ dx #?

# int \ (4x^3-7x)/(x^4-5x^2+4) \ dx = 3/2 ln|x^2-4| + 1/2 ln|x^2-1| + c #

Denote the integral by:

And so the partial fraction decomposition will be of the form:

Then if we put the RHS over a common denominator we end up with the identity:

So using partial fraction decomposition we have:

These are all trivial integrals, so we can integrate to get:

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To evaluate the integral (\int \frac{4x^3 - 7x}{x^4 - 5x^2 + 4} , dx), we first perform polynomial long division to simplify the integrand. After division, we get:

[\frac{4x^3 - 7x}{x^4 - 5x^2 + 4} = \frac{4x^3 - 7x}{(x^2 - 4)(x^2 - 1)}]

Then, we use partial fraction decomposition to express the integrand as a sum of simpler fractions:

[\frac{4x^3 - 7x}{(x^2 - 4)(x^2 - 1)} = \frac{A}{x-2} + \frac{B}{x+2} + \frac{C}{x-1} + \frac{D}{x+1}]

Solve for A, B, C, and D by equating coefficients, then integrate each term separately. After integrating, you'll have:

[\int \frac{4x^3 - 7x}{x^4 - 5x^2 + 4} , dx = \ln|x-2| - \ln|x+2| - \ln|x-1| + \ln|x+1| + C]

Where (C) is the constant of integration.

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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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