Let $k=\displaystyle\int_0^2 x^2(x^2+2)^{1/5}\,dx$. Which of the following is/are INCORRECT?
Step-by-Step Solution
Key Concept: Bound $k$: $x^2(x^2+2)^{1/5}$ is increasing on $[0,2]$. Lower bound: $(2^{1/5})\int_0^2 x^2 dx=2^{1/5}\cdot 8/3\approx 1.149\times 2.667\approx 3.06$. Upper bound: $6^{1/5}\cdot 8/3\approx 1.431\times 2.667\approx 3.82$. So $3<k<4$ (B ✓, meaning B is CORRECT). $[k]=3$ (not 2 — A is INCORRECT ✓). $k$ is not an integer so not prime or odd in integer sense (C,D INCORRECT ✓).
A ✗ ($[k]=3\neq 2$), B ✓ ($3<k<4$), C ✗ (not integer), D ✗ (not integer). INCORRECT statements: A, C, D.
Correct Answer: ACD