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Differentiate an equation with an integral of variable limit (Leibniz rule)

This type of question, asked 2 times: 14 May 2026 S1, 19 May 2026 S1. All 2 questions below, oldest first.

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  1. Question 1

    14 May 2026 S1

    82
    If ∫0xf(t) dt=x+∫x0t⋅f(t) dt\int_0^x f(t)\,dt = x + \int_x^0 t \cdot f(t)\,dt then the value of f(1)=f(1) =

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  2. Question 2

    19 May 2026 S1

    85
    If ∫0f(x)dt=xcos⁡(πx)\displaystyle\int_0^{f(x)} dt = x\cos(\pi x) then f′(4)=…f'(4) = \ldots

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