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Limit at pi/2 of the integral from x^3 to (pi/2)^3 of a trig function of t^(1/3), over (x - pi/2)^2

The same question, asked 2 times: 29 Jan 2024*, 9 Apr 2024. Every wording below, oldest first.

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  1. Question 1

    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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  2. Question 2

    9 Apr 2024

    160
    lim⁡x→π2(∫x3(π/2)3(sin⁡(2t1/3)+cos⁡(t1/3))dt(x−π2)2)\lim_{x \rightarrow\frac{\pi}{2}} \left( \frac{\int_{x^{3}}^{(\pi/2)^{3}} \left( \sin\left( 2t^{1/3} \right)+ \cos\left( t^{1/3} \right) \right)dt}{\left( x -\frac{\pi}{2} \right)^{2}} \right) is equal to:

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