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Odd and even functions over symmetric limits

This type of question, asked 13 times: May 2021 S1, 3 May 2023 S2, 4 May 2023 S1, 15 May 2023 S1, 10 May 2024 S2, 19 Apr 2025 S2, 20 Apr 2025 S1, 25 Apr 2025 S2, 13 Apr 2026 S1, 15 Apr 2026 S2, 16 Apr 2026 S2, 20 Apr 2026 S2, 19 May 2026 S1. All 13 questions below, oldest first.

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  1. Question 1

    May 2021 S1

    123
    The value of ∫−1117x5−x4+29x3−31x+1x2+1 dx\int_{-1}^{1}\frac{17x^5 - x^4 + 29x^3 - 31x + 1}{x^2+1}\,dx

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  2. Question 2

    3 May 2023 S2

    119
    The value of the integral ∫−π/2π/2[x2+log⁡ ⁣(π−xπ+x)]cos⁡x dx\int_{-\pi/2}^{\pi/2}\left[x^{2}+\log\!\left(\frac{\pi-x}{\pi+x}\right)\right]\cos x\,dx is equal to

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  3. Question 3

    4 May 2023 S1

    126
    The integral ∫−1/21/2([x]+log⁡e1+x1−x)dx\int_{-1/2}^{1/2}\left([x]+\log_e\frac{1+x}{1-x}\right)dx, where [x][x] represents greatest integer function, equals

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  4. Question 4

    15 May 2023 S1

    132
    The value of integral ∫−20[x3+3x2+3x+5+(x+1)cos⁡(x+1)]dx\int_{-2}^{0}\left[x^{3}+3x^{2}+3x+5+(x+1)\cos(x+1)\right]dx is equal to

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  5. Question 5

    10 May 2024 S2

    145
    ∫−π/2π/2[f(x)+f(−x)][g(x)−g(−x)] dx\displaystyle\int_{-\pi/2}^{\pi/2}[f(x)+f(-x)][g(x)-g(-x)]\,dx

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  6. Question 6

    19 Apr 2025 S2

    134
    The value of ∫−33sin⁡7xcos⁡16x dx\int_{- 3}^{3} \sin^{7}x\cos^{16}x\text{ }dx is

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  7. Question 7

    20 Apr 2025 S1

    126
    The value of ∫−11(1+x+x2−1−x+x2)dx\int_{- 1}^{1} \left( \sqrt{1 +x+x^{2}}-\sqrt{1 -x+x^{2}} \right)dx is

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  8. Question 8

    25 Apr 2025 S2

    126
    ∫log⁡12log⁡2sin⁡(ex−1ex+1)dx=\int_{\log\frac{1}{2}}^{\log2} \sin\left( \frac{e^{x}- 1}{e^{x}+ 1} \right)dx=

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  9. Question 9

    13 Apr 2026 S1

    81
    The value of the integral ∫−33(2x9+3x8−5x7+9x6+4x3−x+3)x2+3 dx\displaystyle\int_{-\sqrt{3}}^{\sqrt{3}}\dfrac{(2x^9 + 3x^8 - 5x^7 + 9x^6 + 4x^3 - x + 3)}{x^2 + 3}\,dx is

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  10. Question 10

    15 Apr 2026 S2

    82
    If f(x)f(x) is an even function, then ∫−22(∣x∣+f(x)sin⁡x) dx\displaystyle\int_{-2}^{2}(|x| + f(x)\sin x)\,dx is

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  11. Question 11

    16 Apr 2026 S2

    81
    The value of ∫−ππ2x(1+sin⁡x)1+cos⁡2x dx\displaystyle\int_{-\pi}^{\pi}\dfrac{2x(1 + \sin x)}{1 + \cos^2 x}\,dx is...

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  12. Question 12

    20 Apr 2026 S2

    83
    If ∫−π/2π/2(sin⁡2x+sin⁡3x) dx=k\int_{-\pi/2}^{\pi/2}(\sin^2 x + \sin^3 x)\,dx = k, then the value of kk

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  13. Question 13

    19 May 2026 S1

    83
    Let f(x)f(x) be odd and continuous on [−a,a][-a, a]. Then the value of ∫−aaxf(x2)dx\int_{-a}^{a} xf(x^2)dx is:

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