A roaring paper tiger (My sixth integral problem)

Calculus Level 5

\[ \displaystyle \int_{-1}^{1} \sqrt[3]{x^3 + 3x^2 + \sqrt[3]{27x^3 + 27x^2 + \sqrt[3]{729x^3 + 243x^2 +\sqrt[3]{19683x^3 + 2182 x^2 +\cdots} }}}dx \]

Evaluate the integral above.

Clarification:

  • The coefficient of \(x^3 \) is given by \(3^{3n-3} \), where \(n\) is a positive integer.

  • The coefficient of \(x^2 \) is given by \(3^{2n-1} \), where \(n\) is a positive integer.

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