Let $f(x) = x^3 - \frac{3x^2}{2} + x + \frac{1}{4}$. Then the value of $\left[\int_{1/4}^{3/4} f(f(x))dx\right]^{-1}$ is _____.
Step-by-Step Solution
Key Concept: Identify the substitution $t = 2x^{2010}+5x^{804}+10x^{402}$ such that its derivative exactly matches the remaining factors in the integrand.
Substitute $t = 2x^{2010} + 5x^{804} + 10x^{402}$, so $dt = 4020(x^{2009}+x^{803}+x^{401})dx$. The integral becomes $\int\frac{1}{4020}t^{402}dt = \frac{1}{4020} \cdot \frac{t^{403}}{403} + c = \frac{(2x^{2010}+5x^{804}+10x^{402})^{403}}{4020 \cdot 403} + c$. Note that $a = 403 \Rightarrow a - 400 = 3$.
Correct Answer: 4