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Integrals of tan^-1 or cot^-1 of (1 - x + x^2) from 0 to 1

This type of question, asked 3 times: 23 Apr 2025 S1, 16 Apr 2026 S2, 13 May 2026 S2. All 3 questions below, oldest first.

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  1. Question 1

    23 Apr 2025 S1

    121
    The value of ∫01tan⁡−1(1−x+x2)dx\int_{0}^{1} \tan^{- 1}\left( 1 -x+x^{2} \right)dx is

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  2. Question 2

    16 Apr 2026 S2

    80
    The value of integral ∫01cot⁡−1(1+x2−x) dx\displaystyle\int_0^1\cot^{-1}(1 + x^2 - x)\,dx is...

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  3. Question 3

    13 May 2026 S2

    82
    If ∫01tan⁡−1[11−x+x2]dx=2k\int_0^1\tan^{-1}\left[\dfrac{1}{1 - x + x^2}\right]dx = 2k, then the value of kk is:

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