MCQ
For a real number $x$ let $[x]$ denote the largest integer less than or equal to $x$. The smallest positive integer $n$ for which the integral $\int \limits_1^n[x][\sqrt{x}] d x$ exceeds $60$ is
- A$8$
- ✓$9$
- C$10$
- D$\left[60^{2 / 3}\right]$
Let $I=\int \limits_1^n[x][\sqrt{x}] d x$
$\Rightarrow I=\int \limits_1^2 d x+\int \limits_2^3 d x+\int \limits_3^4 3 d x+\int \limits_4^5 8 d x+\int \limits_5^6 10 d x$
$+\int \limits_6^7 12 d x+\int \limits_7^8 14 d x+\int \limits_8^9 16 d x+\int \limits_9^{10} 27 d x+\ldots$
But $I > 60$ $I=1+2+3+8+10+12+14+16=66$
So, least value of $n=9$
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