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Integral of the greatest integer of a trigonometric expression, split at its break points

This type of question, asked 5 times: 2021 Paper 7, 2021 Paper 12, 25 Jul 2022, 8 Apr 2023, 28 Jan 2026 S2. All 5 questions below, oldest first.

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  1. Question 1

    2021 Paper 7

    87
    If [.] represents the greatest integer function, then the value of ∣∫0π2[[x2]−cos⁡x]dx∣\left| \int_{0}^{\sqrt{\frac{\pi}{2}}} \left\lbrack \left\lbrack x^{2} \right\rbrack- \cos x \right\rbrack dx \right| is

  2. Question 2

    2021 Paper 12

    44
    If [x]\lbrack x\rbrack denotes the greatest integer less than or equal to xx, then the value of the integral ∫−π/2π/2[[x]−sin⁡x]dx\int_{-\pi/2}^{\pi/2} \lbrack\lbrack x\rbrack - \sin x\rbrack dx is equal to:

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  3. Question 3

    25 Jul 2022

    159
    Let [t]\lbrack t\rbrack denote the greatest integer less than or equal to tt. Then the value of the integral ∫−3101([sin⁡(πx)]+e[cos⁡(2πx)])dx\int_{- 3}^{101} \left( \lbrack \sin(\pi x)\rbrack +e^{\lbrack \cos(2\pi x)\rbrack} \right)dx is equal to

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  4. Question 4

    8 Apr 2023

    81
    Let [t]\lbrack t\rbrack denotes the greatest integer ≤t\leq t. Then 2π∫π/65π/6(8[csc⁡x]−5[cot⁡x])dx\frac{2}{\pi}\int_{\pi/6}^{5\pi/6} (8\lbrack \csc x\rbrack - 5\lbrack \cot x\rbrack)dx is equal to

  5. Question 5

    28 Jan 2026 S2

    63
    Let [ • ] denote the greatest integer function. Then ∫−π2π2(12(3+[x])3+[sin⁡x]+[cos⁡x])dx\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \left( \frac{12(3 + \lbrack x\rbrack)}{3 + \lbrack \sin x\rbrack + \lbrack \cos x\rbrack} \right)dx is equal to:

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