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Integral over [-a, a] of f(x)/(1 + b^(odd function)): reduces to the integral of f from 0 to a

This type of question, asked 11 times: 2021 Paper 16, 2021 Paper 19, 2021 Paper 25, 14 Jun 2022, 27 Jun 2022, 29 Jan 2024, 1 Feb 2024, 4 Apr 2024, 28 Jan 2025, 2 Apr 2025, 6 Apr 2026 S1. All 11 questions below, oldest first.

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  1. Question 1

    2021 Paper 16

    44
    ∫−π4π4dx(1+excosx)(sin⁡4x+cos⁡4x)\int_{\frac{-\pi}{4}}^{\frac{\pi}{4}} \frac{dx}{\left( 1 +e^{xcosx} \right)\left( \sin^{4}x+\cos^{4}x \right)}

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  2. Question 2

    2021 Paper 19

    78
    The value of ∫−π2π2(1+sin⁡2x1+πsin⁡x)dx\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \left( \frac{1 +\sin^{2}x}{1 +\pi^{\sin x}} \right)dx is

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  3. Question 3

    2021 Paper 25

    75
    The value of ∫−π/2π/2cos⁡2x1+3xdx\int_{-\pi/2}^{\pi/2} \frac{\cos^{2}x}{1 +3^{x}}dx is:

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  4. Question 4

    14 Jun 2022

    156
    The value of the integral ∫−π/2π/2dx(1+ex)(sin⁡6x+cos⁡6x)\int_{-\pi/2}^{\pi/2} \frac{dx}{\left( 1 +e^{x} \right)\left( \sin^{6}x+\cos^{6}x \right)} is equal to

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  5. Question 5

    27 Jun 2022

    70
    The value of the integral ∫−22∣x3+x∣(ex∣x∣+1)dx\int_{- 2}^{2} \frac{\left| x^{3}+ x \right|}{\left( e^{x|x|}+ 1 \right)}dx is equal to :

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  6. Question 6

    29 Jan 2024

    78
    If the value of the integral ∫−π2π2(x2cos⁡x1+πx+1+sin⁡2x1+esin⁡x2023)dx=π4(π+a)−2\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \left( \frac{x^{2}\cos x}{1 +\pi^{x}}+\frac{1 +\sin^{2}x}{1 +e^{\sin x^{2023}}} \right)dx =\frac{\pi}{4}(\pi + a) - 2, then the value of aa is

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  7. Question 7

    1 Feb 2024

    88
    If ∫−π/2π/282cos⁡xdx(1+esin⁡x)(1+sin⁡4x)=απ+βlog⁡e(3+2\int_{- \pi/2}^{\pi/2} \frac{8\sqrt{2}\cos xdx}{\left( 1 +e^{\sin x} \right)\left( 1 +\sin^{4}x \right)}= \alpha\pi + \beta\log_{e}(3 + 2 2\sqrt{2} ), where α,β\alpha,\beta are integers, then α2+β2\alpha^{2}+\beta^{2} equals

  8. Question 8

    4 Apr 2024

    161
    If the value of the integral ∫−11cos⁡αx1+3xdx\int_{- 1}^{1} \frac{\cos\alpha x}{1 +3^{x}}dx is 2π\frac{2}{\pi}. Then, a value of α\alpha is

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  9. Question 9

    28 Jan 2025

    63
    If ∫−π2π296x2cos⁡2x(1+ex)dx=π(απ2+β),α,β∈Z\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \frac{96x^{2}\cos^{2}x}{\left( 1 +e^{x} \right)}dx = \pi\left( \alpha\pi^{2}+ \beta \right),\alpha,\beta \in Z, then (α+β)2(\alpha + \beta)^{2} equals :

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  10. Question 10

    2 Apr 2025

    132
    Let (a,b)\left( a,b \right) be the point of intersection of the curve x2=2yx^{2}= 2y and the straight line y−y- 2x−6=02x- 6 = 0 in the second quadrant. Then the integral I=∫ab9x21+5xdxI=\int_{a}^{b} \frac{9x^{2}}{1 +5^{x}}dx is equal to:

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  11. Question 11

    6 Apr 2026 S1

    68
    The value of the integral ∫−π4π4(32cos⁡4x1+esin⁡x)dx\int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} \left( \frac{32\cos^{4}x}{1 +e^{\sin x}} \right)dx is :

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