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Integrals with the fractional part x - [x], using its period 1

This type of question, asked 6 times: 2021 Paper 6, 2021 Paper 8, 2021 Paper 11, 2021 Paper 13, 2021 Paper 19, 2021 Paper 25. All 6 questions below, oldest first.

Try each one first, then open its answer.

  1. Question 1

    2021 Paper 6

    65
    Consider the integral
    I=∫010[x]e[x]ex−1dxI=\int_{0}^{10} \frac{\lbrack x\rbrack e^{\lbrack x\rbrack}}{e^{x- 1}}dx
    where [x]\lbrack x\rbrack denotes the greatest integer less than or equal to xx. Then the value of II is equal to:

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  2. Question 2

    2021 Paper 8

    62
    If the integral ∫010[sin⁡2πx]ex−[x]dx=αe−1+βe−12+γ\int_{0}^{10} \frac{\lbrack \sin2\pi x\rbrack}{e^{x- \lbrack x\rbrack}}dx=\alpha e^{- 1}+\beta e^{-\frac{1}{2}}+\gamma, where α,β,γ\alpha,\beta,\gamma are integers and [x]\lbrack x\rbrack denotes the greatest integer less than or equal to xx, then the value of α+β+γ\alpha+\beta+\gamma is equal to:

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  3. Question 3

    2021 Paper 11

    43
    Let aa be a positive real number such that ∫0aex−[x]dx=10e−9\int_{0}^{a} e^{x- \lbrack x\rbrack}dx= 10e- 9 where [x]\lbrack x\rbrack is the greatest integer less than or equal to xx. Then a is equal to:

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  4. Question 4

    2021 Paper 13

    59
    ∫0100πsin⁡2xe(xπ−[xπ])dx=απ31+4π2,α∈R\int_{0}^{100\pi} \frac{\sin^{2}x}{e^{\left( \frac{x}{\pi}-\left\lbrack \frac{x}{\pi} \right\rbrack \right)}}dx=\frac{\alpha\pi^{3}}{1 + 4\pi^{2}},\alpha\in R, where [x]\lbrack x\rbrack is the greatest integer function, then α\alpha is

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  5. Question 5

    2021 Paper 19

    64
    If the value of the integral ∫05x+[x]ex−[x]dx=αe−1+β\int_{0}^{5} \frac{x + \lbrack x\rbrack}{e^{x - \lbrack x\rbrack}}dx =\alpha e^{- 1}+\beta, where α,β∈R,5α+6β=0\alpha,\beta\in R,5\alpha+ 6\beta= 0, and [x]\lbrack x\rbrack denotes the greatest integer less than or equal to xx; then the value of (α+β)2(\alpha+\beta)^{2} is equal to:

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  6. Question 6

    2021 Paper 25

    63
    The value of ∑n=1100∫n−1nex−[x]dx\sum_{n= 1}^{100} \int_{n- 1}^{n} e^{x- \lbrack x\rbrack}dx, where [x]\lbrack x\rbrack is the greatest integer ≤x\leq x, is:

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