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Integral of f/(f + g) where x -> a + b - x swaps f and g: half the interval

This type of question, asked 6 times: 2021 Paper 7, 2021 Paper 14, 2021 Paper 20, 24 Jan 2023, 23 Jan 2025, 23 Jan 2026 S1. All 6 questions below, oldest first.

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  1. Question 1

    2021 Paper 7

    78
    Which of the following statements is incorrect for the function g(α)g(\alpha) for α∈R\alpha\in R such that g(α)=∫π6π3sin⁡αxcos⁡αx+sin⁡αxdxg(\alpha) =\int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \frac{\sin^{\alpha}x}{\cos^{\alpha}x+\sin^{\alpha}x}dx

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  2. Question 2

    2021 Paper 14

    46
    I=∫π245π24dx1+tan⁡2x3I=\int_{\frac{\pi}{24}}^{\frac{5\pi}{24}} \frac{dx}{1 +\sqrt[3]{\tan2x}} is equal to

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  3. Question 3

    2021 Paper 20

    79
    ∫616log⁡ex2log⁡ex2+log⁡e(x2−44x+484)dx\int_{6}^{16} \frac{\log_{e}x^{2}}{\log_{e}x^{2}+\log_{e}\left( x^{2}- 44x+ 484 \right)}dx is equal to:

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  4. Question 4

    24 Jan 2023

    85
    The value of 8π∫0π2(cos⁡x)2023(sin⁡x)2023+(cos⁡x)2023dx\frac{8}{\pi}\int_{0}^{\frac{\pi}{2}} \frac{(\cos x)^{2023}}{(\sin x)^{2023}+ (\cos x)^{2023}}dx is

  5. Question 5

    23 Jan 2025

    51
    The value of ∫e2e41x(e((log⁡ex)2+1)−1e((log⁡ex)2+1)−1+e((6−log⁡ex)2+1)−1)dx\int_{e^{2}}^{e^{4}} \frac{1}{x}\left(\frac{e^{\left(\left(\log_{e}x\right)^{2}+1\right)^{-1}}}{e^{\left(\left(\log_{e}x\right)^{2}+1\right)^{-1}}+e^{\left(\left(6-\log_{e}x\right)^{2}+1\right)^{-1}}}\right)dx is

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  6. Question 6

    23 Jan 2026 S1

    60
    The value of the integral ∫π245π24dx1+tan⁡2x3\int_{\frac{\pi}{24}}^{\frac{5\pi}{24}} \frac{dx}{1 +\sqrt[3]{\tan2x}} is :

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