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Integral of f(x)/(1 + a^x) over symmetric limits (or over 0 to pi with a^cos x)

This type of question, asked 5 times: 9 May 2023 S1, 16 May 2023 S1, 15 Apr 2026 S1, 17 Apr 2026 S1, 12 May 2026 S2. All 5 questions below, oldest first.

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  1. Question 1

    9 May 2023 S1

    136
    The value of I=∫−π/2π/2x2cos⁡x1+e−x dxI = \int_{-\pi/2}^{\pi/2}\frac{x^2\cos x}{1+e^{-x}}\,dx is equal to

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  2. Question 2

    16 May 2023 S1

    122
    The value of ∫−π/2π/2sin⁡2x1+2x dx\int_{-\pi/2}^{\pi/2} \frac{\sin^2 x}{1+2^x}\,dx is

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  3. Question 3

    15 Apr 2026 S1

    82
    The value of the integral ∫−π/2π/2(x2cos⁡x1+ex)dx\displaystyle\int_{-\pi/2}^{\pi/2}\left(\dfrac{x^2\cos x}{1 + e^x}\right)dx is equal to (π2A)−B\left(\dfrac{\pi^2}{A}\right) - B. Then (AB)=\left(\dfrac{A}{B}\right) =

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  4. Question 4

    17 Apr 2026 S1

    83
    The value of ∫0π11+2cos⁡x dx\int_0^\pi\dfrac{1}{1 + 2^{\cos x}}\,dx is ____

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  5. Question 5

    12 May 2026 S2

    82
    The value of the integral ∫−111(1+ex)(1+x2) dx\int_{-1}^{1}\dfrac{1}{(1 + e^x)(1 + x^2)}\,dx is....

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