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Limit involving an integral with a variable limit

This type of question, asked 6 times: 2021 Paper 1, 2021 Paper 24, 30 Jan 2023, 29 Jan 2024, 30 Jan 2024, 4 Apr 2024. All 7 questions below, oldest first.

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  1. Question 1

    2021 Paper 1

    70
    lim⁡x→0∫0x2(sin⁡t)dtx3\lim_{x\rightarrow 0} \frac{\int_{0}^{x^{2}} (\sin\sqrt{t})dt}{x^{3}} is equal to:

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  2. Question 2

    2021 Paper 24

    61
    Let f:R→Rf:R \rightarrow R be a continuous function. Then lim⁡x→π4π4∫2sec⁡2xf(x)dxx2−π216\lim_{x\rightarrow\frac{\pi}{4}} \frac{\frac{\pi}{4}\int_{2}^{\sec^{2}x} f(x)dx}{x^{2}-\frac{\pi^{2}}{16}} is equal to:

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  3. Question 3

    30 Jan 2023

    90
    lim⁡x→048x4∫0xt3t6+1dt\lim_{x \rightarrow 0} \frac{48}{x^{4}}\int_{0}^{x} \frac{t^{3}}{t^{6}+ 1}dt is equal to

  4. Question 4

    29 Jan 2024

    66
    lim⁡x→π2(1(x−π2)2∫x3(π2)3cos⁡(t13)dt)\lim_{x \rightarrow\frac{\pi}{2}} \left( \frac{1}{\left( x -\frac{\pi}{2} \right)^{2}}\int_{x^{3}}^{\left( \frac{\pi}{2} \right)^{3}} \cos\left( t^{\frac{1}{3}} \right)dt \right) is equal to

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  5. Question 5

    29 Jan 2024

    180
    Let the slope of the line 45x+5y+3=045x+ 5y+ 3 = 0 be 27r1+9r2227r_{1}+\frac{9r_{2}}{2} for some  r1, r2∈R\ r_{1},\ r_{2}\in R. Then Limx→3(∫3x8t23r2x2−r2x2−r1x3−3xdt)Lim_{x\rightarrow 3}\left( \int_{3}^{x} \frac{8t^{2}}{\frac{3r_{2}x}{2}-r_{2}x^{2}-r_{1}x^{3}- 3x}dt \right) is equal to

  6. Question 6

    30 Jan 2024

    80
    Let f:[−π2,π2]→Rf:\left\lbrack -\frac{\pi}{2},\frac{\pi}{2} \right\rbrack\rightarrow R be a differentiable function such that f(0)=12f(0) =\frac{1}{2}, If the lim⁡x→0x∫0xf(t)dtex2−1=α\lim_{x \rightarrow 0} \frac{x\int_{0}^{x} f(t)dt}{e^{x^{2}}- 1}= \alpha, then 8α28\alpha^{2} is equal to :

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  7. Question 7

    4 Apr 2024

    158
    Let f(x)=∫0x(t+sin⁡(1−et))dt,x∈Rf(x) =\int_{0}^{x} \left( t + \sin\left( 1 -e^{t} \right) \right)dt,x \in R. Then lim⁡x→0f(x)x3\lim_{x \rightarrow 0} \frac{f(x)}{x^{3}} is equal to

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