The value of $2^{2010} \frac{\int_0^1 x^{1004}(1-x)^{1004} dx}{\int_0^1 x^{1004}\left(1-x^{2010}\right)^{1004} dx}$ is ____.
Step-by-Step Solution
Key Concept: Use substitution and symmetry properties to relate two integrals and compute their ratio by careful manipulation of bounds.
Split $I_1 = ∫_0^1x^{1004}(1-x)^{1004}dx = 2∫_0^{1/2}x^{1004}(1-x)^{1004}dx$ using symmetry. For $I_2 = ∫_0^1x^{1004}(1-2010)^{1004}dx$, substitute $u = 1005x$ to get $I_2 = \frac{1}{1005}∫_0^{1005}(1-t)^{1004}t^{1004}dt$. Through substitution $t = 2y$ and integration, establish that $\frac{I_1}{I_2} = \frac{2^{2010}}{2^{2008}} = 4$.
Correct Answer: 4