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Differentiating an integral with a variable limit to find f

This type of question, asked 4 times: 27 Jun 2022, 10 Apr 2023, 1 Feb 2024, 28 Jan 2025. All 4 questions below, oldest first.

Try each one first, then open its answer.

  1. Question 1

    27 Jun 2022

    158
    Let ff be a differentiable function in (0,π2)\left( 0,\frac{\pi}{2} \right). If ∫cos⁡x1t2f(t)dt=sin⁡3x+cos⁡x\int_{\cos x}^{1} t^{2}f(t)dt =\sin^{3}x + \cos x then 13f′(13)\frac{1}{\sqrt{3}}f^{'}\left( \frac{1}{\sqrt{3}} \right) is equal to:

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  2. Question 2

    10 Apr 2023

    151
    Let ff be aa continuous function satisfying ∫0t2(f(x)+x2)dx=43t3,∀t>0\int_{0}^{t^{2}} \left( f(x) +x^{2} \right)dx =\frac{4}{3}t^{3},\forall t > 0. Then f(π24)f\left( \frac{\pi^{2}}{4} \right) is equal to:

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  3. Question 3

    1 Feb 2024

    173
    Let f:(0,∞)→Rf:(0,\infty) \rightarrow R and F(x)=∫0xtf(t)dtF(x) =\int_{0}^{x} tf(t)dt. If F(x2)=F\left( x^{2} \right)= x4+x5x^{4}+x^{5}, then ∑r=112f(r2)\sum_{r = 1}^{12} f\left( r^{2} \right) is equal to:

  4. Question 4

    28 Jan 2025

    132
    Let ff be a real valued continuous function defined on the positive real axis such that g(x)=∫0xtf(t)dtg(x) =\int_{0}^{x} tf(t)dt. If g(x3)=x6+x7g\left( x^{3} \right)=x^{6}+x^{7}, then value of ∑r=115f(r3)\sum_{r = 1}^{15} f\left( r^{3} \right) is :

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