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Equation with an integral of f up to x: differentiate to get a differential equation

This type of question, asked 8 times: 2021 Paper 26, 31 Jan 2023, 3 Apr 2025, 4 Apr 2025, 22 Jan 2026 S1, 23 Jan 2026 S1, 5 Apr 2026 S2, 6 Apr 2026 S2. All 8 questions below, oldest first.

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  1. Question 1

    2021 Paper 26

    79
    Let f(x)=∫0xetf(t)dt+exf(x) =\int_{0}^{x} e^{t}f(t)dt +e^{x} be a differentiable function for all x∈Rx \in R. Then f(x)f(x) equals.

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  2. Question 2

    31 Jan 2023

    62
    Let a differentiable function ff satisfy f(x)+∫3xf(t)tdt=x+1,x≥3f(x) +\int_{3}^{x} \frac{f(t)}{t}dt =\sqrt{x + 1},x \geq 3. Then 12f(8)12f(8) is equal to:

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  3. Question 3

    3 Apr 2025

    62
    Let gg be a differentiable function such that ∫0xg(t)dt=x−∫0xtg(t)dt,x≥0\int_{0}^{x} g(t)dt = x -\int_{0}^{x} tg(t)dt,x \geq 0 and let y=y(x)y = y(x) satisfy the differential equation dydx−ytanx=\frac{dy}{dx}-ytanx= 2(x+1)secxg(x),x∈[0,π2)2(x+ 1)secxg(x),x\in\left\lbrack 0,\frac{\pi}{2} \right). If y(0)=0y(0) = 0, then y(π3)y\left( \frac{\pi}{3} \right) is equal to.

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  4. Question 4

    4 Apr 2025

    58
    Let f:[0,∞]→Rf:\left\lbrack 0,\infty \right\rbrack\rightarrow\mathbb{R} be differentiable function such that f(x)=1−2x+f\left( x \right)= 1 - 2x + ∫0xex−tf(t)dt\int_{0}^{x} e^{x-t}f(t)dt for all x∈[0,∞)x \in \lbrack 0,\infty).Then the area of the region bounded by y=f(x)y =f(x) and the coordinate axes is

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  5. Question 5

    22 Jan 2026 S1

    59
    Let f:[1,∞)→Rf:\lbrack 1,\infty) \rightarrow R be a differentiable function, If 6∫1xf(t)dt=3xf(x)+x3−46\int_{1}^{x} f(t)dt = 3xf(x) +x^{3}- 4 for all x≥1x \geq 1, then the value of f(2)−f(3)f(2) - f(3) is

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  6. Question 6

    23 Jan 2026 S1

    73
    Let f be a twice differentiable non-negative function such that (f(x))2=25+∫0x((f(t))2+(f′(t))2)dt(f(x))^{2} = 25 + \int_{0}^{x} \left( (f(t))^{2} + \left( f'(t) \right)^{2} \right)dt. Then the mean of f(log⁡e(1)),f(log⁡e(2)),…..,f(log⁡e(625))f\left( \log_{e}(1) \right),f\left( \log_{e}(2) \right),\ldots..,f\left( \log_{e}(625) \right) is equal to ____\_\_\_\_.

  7. Question 7

    5 Apr 2026 S2

    69
    Let f:[1,∞)→Rf:\lbrack 1,\infty) \rightarrow R be a differentiable function defined as f(x)=∫1xf(t)dt+(1−x)(log⁡ex−1)+ef(x) =\int_{1}^{x} f(t)dt+ (1 -x)\left( \log_{e}x- 1 \right)+e. Then the value of f(f(1))f(f(1)) is :

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  8. Question 8

    6 Apr 2026 S2

    68
    Let f:R→Rf:R\rightarrow R be such that f(xy)=f(x)f(y)f(xy) =f(x)f(y), for all x,y∈Rx,y\in R and f(0)≠0f(0) \neq 0. Let g:[1,∞)→Rg:\lbrack 1,\infty) \rightarrow R be a differentiable function such that x2g(x)=∫1x(t2f(t)−tg(t))dtx^{2}g(x) =\int_{1}^{x} \left( t^{2}f(t) - tg(t) \right)dtThen g(2)g(2) is equal to :

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