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

by RoRi

December 3, 2015

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

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

Stumbling Robot

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

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