Three 3's

Calculus Level 4

\[ \begin{eqnarray} \displaystyle & \int_0^1 & (1-x^3)^3 \ dx \\ \displaystyle & \int_0^1 & x^3 (1-x^3)^3 \ dx \\ \end{eqnarray} \]

Denote the values of the two definite integrals above as \(A \) and \( \frac A k \) respectively. Find the value of \(k\).

Bonus: Try solving this question without finding the value of \(A\).

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