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Integral of log(t + sqrt(1 + t^2))/sqrt(1 + t^2) written as (1/2)(g(t))^2: find g(2)

The same question, asked 2 times: 16 May 2023 S1, 13 May 2024 S2. Every wording below, oldest first.

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  1. Question 1

    16 May 2023 S1

    149
    If ∫log⁡ ⁣(t+1+t2)1+t2 dt=12(g(t))2+c\int\frac{\log\!\left(t+\sqrt{1+t^2}\right)}{\sqrt{1+t^2}}\,dt = \frac{1}{2}(g(t))^2 + c where c is a constant of integration, then g(2)g(2) is equal to

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  2. Question 2

    13 May 2024 S2

    129
    If ∫log⁡(t+1+t2)1+t2 dt=12[g(t)]2+c\int\frac{\log(t+\sqrt{1+t^2})}{\sqrt{1+t^2}}\,dt=\frac{1}{2}[g(t)]^2+c, (Where cc is a constant of integration) then g(2)g(2) is:

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