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Integral involving the greatest integer or fractional part function: split where it jumps

This type of question, asked 8 times: May 2022 S1, 11 May 2023 S1, 19 Apr 2025 S1, 21 Apr 2025 S1, 17 Apr 2026 S2, 18 Apr 2026 S1, 19 Apr 2026 S1, 12 May 2026 S2. All 9 questions below, oldest first.

Try each one first, then open its answer.

  1. Question 1

    May 2022 S1

    141
    ∫02[x] dx+∫02∣x−1∣ dx=\int_0^2 [x]\,dx + \int_0^2 |x-1|\,dx = (where [x][x] denotes the greatest integer function.)

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  2. Question 2

    11 May 2023 S1

    111
    ∫0.23.5[x] dx=\int_{0.2}^{3.5} [x]\,dx = (where [x]=[x] = greatest integer not greater than x)

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  3. Question 3

    19 Apr 2025 S1

    113
    The value of ∫14log⁡[x]dx\int_{1}^{4} \log\lbrack x\rbrack dx, where [x]\lbrack x\rbrack is the greatest integer function less than or equal to xx is equal to

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  4. Question 4

    21 Apr 2025 S1

    116
    The value of ∫02[x2]dx\int_{0}^{2} \left\lbrack x^{2} \right\rbrack dx is (where [x]\lbrack x\rbrack denotes the greatest integer function not greater than xx )

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  5. Question 5

    17 Apr 2026 S2

    84
    Let f(x)=x−[x]f(x) = x - [x] for every real number xx, where [x][x] is integral part of xx. then ∫−11f(x) dx\displaystyle\int_{-1}^{1} f(x)\,dx is

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  6. Question 6

    18 Apr 2026 S1

    83
    If [⋅][\cdot] denotes the greatest integer function, then ∫0π/3[tan⁡x] dx=\int_0^{\pi/3}[\tan x]\,dx =

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  7. Question 7

    19 Apr 2026 S1

    83
    The value of ∫24({x}+[x]) dx=\int_2^4(\{x\} + [x])\,dx =...(where {x}\{x\} and [x][x] are the fractional part function and the greatest integer function, respectively)

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  8. Question 8

    19 Apr 2026 S1

    82
    If [x][x] is the greatest integer function not greater than xx, then the value of ∫02x[x2] dx\int_0^2 x[x^2]\,dx is...

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  9. Question 9

    12 May 2026 S2

    83
    If f(x)={[x]⋅∣x∣x,x≠00,x=0f(x) = \begin{cases} \dfrac{[x] \cdot |x|}{x}, & x \neq 0 \\ 0, & x = 0 \end{cases}, where [x][x] and ∣x∣|x| denote the greatest integer function and the modulus function respectively, then ∫−21f(x) dx=…\int_{-2}^{1} f(x)\,dx = \ldots

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