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Integral of the greatest integer of a polynomial, split where it crosses integers

This type of question, asked 4 times: 2021 Paper 2, 28 Jun 2022, 8 Apr 2023, 6 Apr 2024. All 4 questions below, oldest first.

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  1. Question 1

    2021 Paper 2

    68
    The value of the integral, ∫13[x2−2x−2]dx\int_{1}^{3} \left\lbrack x^{2}- 2x- 2 \right\rbrack dx, where [x]\lbrack x\rbrack denotes the greatest integer less than or equal to xx, is:

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  2. Question 2

    28 Jun 2022

    67
    Let [t]\lbrack t\rbrack denote the greatest integer less than or equal to tt. Then, the value of the integral ∫01[−8x2+6x−1]dx\int_{0}^{1} \left\lbrack - 8x^{2}+ 6x - 1 \right\rbrack dx is equal to

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  3. Question 3

    8 Apr 2023

    171
    Let [t] denote the greatest integer function. If ∫02.4[x2]dx=α+β2+γ3+δ5\int_{0}^{2.4} \left\lbrack x^{2} \right\rbrack dx = \alpha + \beta\sqrt{2}+ \gamma\sqrt{3}+ \delta\sqrt{5}, then α+β+γ+\alpha + \beta + \gamma + δ\delta is equal to

  4. Question 4

    6 Apr 2024

    174
    Let [t]\lbrack t\rbrack denote the largest integer less than or equal to tt. If ∫03([x2]+[x22])dx=a+b2−3−5+c6−7\int_{0}^{3} \left( \left\lbrack x^{2} \right\rbrack+\left\lbrack \frac{x^{2}}{2} \right\rbrack \right)dx=a+b\sqrt{2}-\sqrt{3}-\sqrt{5}+c\sqrt{6}-\sqrt{7}, where a,b,c∈za,b,c \in z, then a+b+ca + b + c is equal to