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Using the substitution x -> k/x to evaluate an integral

This type of question, asked 4 times: 2021 Paper 26, 29 Jul 2022, 29 Jan 2023, 5 Apr 2026 S1. All 4 questions below, oldest first.

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  1. Question 1

    2021 Paper 26

    73
    For x>0x > 0, if f(x)=∫1xlog⁡et(1+t)dtf(x) =\int_{1}^{x} \frac{\log_{e}t}{(1 + t)}dt, then f(e)+f(1e)f(e) + f\left( \frac{1}{e} \right) is equal to:

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  2. Question 2

    29 Jul 2022

    75
    If f(α)=∫1αlog⁡10t1+tdt,α>0f(\alpha) =\int_{1}^{\alpha} \frac{\log_{10}t}{1 + t}dt,\alpha > 0, then f(e3)+f(e−3)f\left( e^{3} \right)+ f\left( e^{- 3} \right) is equal to :

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  3. Question 3

    29 Jan 2023

    159
    The value of the integral ∫1/22tan⁡−1xxdx\int_{1/2}^{2} \frac{\tan^{- 1}x}{x}dx is equal to

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  4. Question 4

    5 Apr 2026 S1

    68
    The value of the integral ∫0∞log⁡e(x)x2+4dx\int_{0}^{\infty} \frac{\log_{e}(x)}{x^{2}+ 4}dx is:

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