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Evaluate the integral of (4x2 + x + 1) / (x3 – 1)

Compute the following integral.

    \[ \int \frac{4x^2 + x + 1}{x^3 - 1} \, dx. \]

We can do this without partial fractions, noting that x^3 - 1 = (x-1)(x^2+x+1) we compute

    \begin{align*}  \int \frac{4x^2 + x + 1}{x^3-1} \, dx &= \int \frac{3x^2 + x^2 + x + 1}{x^3-1} \, dx \\  &= \int \frac{3x^2}{x^3-1} \, dx + \int \frac{x^2+x+1}{(x-1)(x^2+x+1)} \, dx \\  &= \log | x^3 - 1| + \log |x-1| + C \\  &= \log \Big( (x-1)^2 (x^2+x+1) \Big)  + C \\  &= 2 \log |x-1| + \log (x^2+x+1) + C. \end{align*}

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