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How can I evaluate the following integral?

$$ \mathcal{I} = \int_0^1 \frac{1}{(1 - x + x^2)\sqrt[3]{x(1 - x)}} dx $$

Notes to the readers

  1. As a 12th grader, this form of integral is pretty much new and odd to me, and I'm not sure how to simplify this. From my math knowledge, substitution, integration by parts, partial fractions, king property, etc. — do not work.

  2. A scientific calculator or integral-calculator.com does not work.

  3. If not a complete solution, consider providing me a clue about how to do this. I want to try it for myself.

  4. The final answer is $\frac{4 \pi}{3\sqrt{3}} \approx 2.4184$.

Rohan Bari
  • 379
  • You can probably contour integrate this with a dogbone contour but given that the final value is pretty nice complex approaches may be cumbersome as there is probably a more elegant real approach. – Max0815 Feb 13 '23 at 16:30
  • @Max0815 Looking at the duplicate post, it seems that complex methods are much, much more efficient here. See Pisco's answer – FShrike Feb 13 '23 at 20:57

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