If [x] represents the greatest integer not exceeding x, then \(\rm \i

1.	If [x] represents the greatest integer not exceeding x, then \(\rm \int_0^9 [x]\ dx\) is1. 322. 363. 404. 28
Answer» Correct Answer - Option 2 : 36 Concept: Definite Integral: If ∫ f(x) dx = g(x) + C, then \(\rm\int_a^b f(x)\ dx= [ g(x)]_a^b\) = g(b) - g(a). \(\rm \int_a^bf(x)\ dx=\int_a^cf(x)\ dx+\int_c^bf(x)\ dx\). \(\rm \sum n=\frac{n(n+1)}{2}\). Calculation: [x] is a multi-valued function in the interval 0-9, with: [x] = 0, x ∈ (0, 1) [x] = 1, x ∈ (1, 2) [x] = 2, x ∈ (2, 3) And so on. ∴ \(\rm \int_0^9 [x]\ dx\) = \(\rm \int_0^1 0\ dx+\int_1^2 1\ dx\ +\ ...\ +\int_8^9 8\ dx\) = \(0[x]_0^1+1[x]_1^2\ +\ ...\ +8[x]_8^9\) = 0[1 - 0] + 1[2 - 1] + ... + 8[9 - 8] = 0 + 1 + 2 + ... + 8 = \(\frac{8(8+1)}{2}\) = 36.

If [x] represents the greatest integer not exceeding x, then \(\rm \int_0^9 [x]\ dx\) is1. 322. 363. 404. 28

Answer» Correct Answer - Option 2 : 36

Concept:

Definite Integral:

If ∫ f(x) dx = g(x) + C, then \(\rm\int_a^b f(x)\ dx= [ g(x)]_a^b\) = g(b) - g(a).

\(\rm \int_a^bf(x)\ dx=\int_a^cf(x)\ dx+\int_c^bf(x)\ dx\).
\(\rm \sum n=\frac{n(n+1)}{2}\).

Calculation:

[x] is a multi-valued function in the interval 0-9, with:

[x] = 0, x ∈ (0, 1)

[x] = 1, x ∈ (1, 2)

[x] = 2, x ∈ (2, 3)

And so on.

∴ \(\rm \int_0^9 [x]\ dx\)

= \(\rm \int_0^1 0\ dx+\int_1^2 1\ dx\ +\ ...\ +\int_8^9 8\ dx\)

= \(0[x]_0^1+1[x]_1^2\ +\ ...\ +8[x]_8^9\)

= 0[1 - 0] + 1[2 - 1] + ... + 8[9 - 8]

= 0 + 1 + 2 + ... + 8

= \(\frac{8(8+1)}{2}\)

= 36.

Discussion

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