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f(x) = g(x) + an integral over fixed limits: treat the integrals as constants

This type of question, asked 5 times: 2021 Paper 24, 14 Jun 2022, 29 Jun 2022, 29 Jan 2023, 24 Jan 2026 S2. All 5 questions below, oldest first.

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  1. Question 1

    2021 Paper 24

    79
    The function f(x)f(x), that satisfies the condition f(x)=x+∫0π/2sin⁡x⋅cos⁡yf(y)dyf(x) =x+\int_{0}^{\pi/2} \sin x\cdot \cos yf(y)dy, is :

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  2. Question 2

    14 Jun 2022

    86
    Let f(θ)=sin⁡θ+∫−π/2π/2(sin⁡θ+tcosθ)f(t)dtf(\theta) = \sin\theta+\int_{-\pi/2}^{\pi/2} (\sin\theta+tcos\theta)f(t)dt. Then the value of ∣∫0π/2f(θ)dθ∣\left| \int_{0}^{\pi/2} f(\theta)d\theta \right| is

  3. Question 3

    29 Jun 2022

    157
    Let ff be a real valued continuous function on [0,1]\lbrack 0,1\rbrack and f(x)=x+∫01(x−t)f(t)dtf(x) = x +\int_{0}^{1} (x - t)f(t)dt. Then which of the following points (x,y)(x,y) lies on the curve y=f(x)y = f(x)?

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  4. Question 4

    29 Jan 2023

    78
    Let f(x)=x+aπ2−4sin⁡x+bπ2−4cos⁡x,x∈Rf(x) = x +\frac{a}{\pi^{2}- 4}\sin x +\frac{b}{\pi^{2}- 4}\cos x,x \in R be a function which satisfies f(x)=x+∫0π/2sin⁡(x+f(x) = x +\int_{0}^{\pi/2} \sin(x + y)f(y)dyy)f(y)dy. Then (a+b)(a + b) is equal to

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  5. Question 5

    24 Jan 2026 S2

    71
    If f(x)f(x) satisfies the relation f(x)=ex+∫01(y+xex)f(y)dyf(x) = e^{x} + \int_{0}^{1} \left( y + {xe}^{x} \right)f(y)dy, then e+f(0)e + f(0) is equal to ____\_\_\_\_ .