Q100 / 180 · KCET 2022 · Mathematics

If $[x]$ is the greatest integer function not greater than $x$, then $\int_0^8[x] d x$ is equal to

  • Correct answer
    A
    28
  • B
    30
  • C
    29
  • D
    20

Solution

Let $I=\int_0^8[x] d x$ $$ \begin{aligned} & I=\int_0^1[x] d x+\int_1^2[x] d x+\int_2^3[x] d x+\int_3^4[x] d x+\int_4^5[x] d x \\ & +\int_5^6[x] d x+\int_6^7[x] d x+\int_7^8[x] d x \\ & =\int_0^1 0 d x+\int_1^2 1 d x+\int_2^3 2 d x+\int_3^4 3 d x+\int_4^5 4 d x \\ & +\int_5^6 5 d x+\int_6^7 6 d x+\int_7^8 7 d x \\ & =0+1[x]_1^2+2[x]_2^3+3[x]_3^4+4[x]_4^5 \\ & +5[x]_5^6+6[x]_6^7+7[x]_7^8 \\ & =0+1(2-1)+2(3-2)+3(4-3) \\ & +4(5-4)+5(6-5)+6(7-6)+7(8-7) \\ & =1+2+3+4+5+6+7 \\ & =\frac{7 \cdot(7+1)}{2}=\frac{7 \cdot 8}{2}=7 \cdot 4=28 \text {. } \\ & \end{aligned} $$

Hence, the correct answer is option: 'A'.

Relevant Questions