Explanation:
(A) : Given,
\(I=\int_{2}^{5} 2[x] \cdot d x\)
\(=\int_{2}^{3} 2[x] d x+\int_{3}^{4} 2[x] d x+\int_{4}^{5} 2[x] d x\)
For, \(\quad 2 \leq x\lt 3,[x]=2\)
For, \(\quad 3 \leq x\lt 4,[x]=3\)
For, \(\quad 4 \leq x\lt 5,[x]=4\)
\(\therefore \quad \mathrm{I}=2 \int_{2}^{3} 2 \mathrm{dx}+2 \int_{3}^{4} 3 \cdot \mathrm{dx}+2 \int_{4}^{5} 4 \cdot \mathrm{dx}\)
\(=2 \times 2(\mathrm{x})_{2}^{3}+6(\mathrm{x})_{3}^{4}+8(\mathrm{x})_{4}^{5}\)
\(=2 \times 2(3-2)+6(4-3)+8(5-4)\)
\(=4+6+8=18\)