# How do you find the integral of #(x^4+x-4) / (x^2+2)#?

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To find the integral of (\frac{{x^4 + x - 4}}{{x^2 + 2}}), you can use polynomial long division to divide (x^4 + x - 4) by (x^2 + 2), then integrate each resulting term. The division results in (x^2 - 2) with a remainder of (x + 2). So the integral becomes:

(\int \frac{{x^4 + x - 4}}{{x^2 + 2}} dx = \int (x^2 - 2) + \frac{{x + 2}}{{x^2 + 2}} dx)

The integral of (x^2 - 2) is straightforward: (\frac{{x^3}}{3} - 2x).

For the fraction (\frac{{x + 2}}{{x^2 + 2}}), perform partial fraction decomposition:

(\frac{{x + 2}}{{x^2 + 2}} = \frac{A}{x - \sqrt{2}} + \frac{B}{x + \sqrt{2}})

Solve for A and B. Then integrate each term separately.

Once integrated, combine the results to get the final integral.

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