_ x → (π)/(2) ((1)/( ( x - (π)/(2))2) ∫ _ x3 ^ ((π)/(2))3...

Let, $y = \lim_{x \to \dfrac{\pi}{2}}\dfrac{{∫}_{{x}^{3}}^{{\left(\dfrac{\pi}{2}\right)}^{3}}cos{t}^{\dfrac{1}{3}}dt}{{\left(x - \dfrac{\pi}{2}\right)}^{2}}$

$⇒ y = \lim_{h \to 0}\dfrac{{∫}_{{\left(\dfrac{\pi}{2} - h\right)}^{3}}^{{\left(\dfrac{\pi}{2}\right)}^{3}}cos{t}^{\dfrac{1}{3}}dt}{{\left(\dfrac{\pi}{2} - h - \dfrac{\pi}{2}\right)}^{2}}$

$⇒ y = \lim_{h \to 0}\left(\dfrac{{∫}_{{\left(\dfrac{\pi}{2} - h\right)}^{3}}^{{\left(\dfrac{\pi}{2}\right)}^{3}}cos{t}^{\dfrac{1}{3}}dt}{{h}^{2}}\right)$

Applying L-Hospital's rule and Newton Leibnitz Theorem we get,

$⇒ y = \lim_{h \to 0}\left(\dfrac{0 + \cos \left(\dfrac{\pi}{2} - h\right) \times 3 \times {\left(\dfrac{\pi}{2} - h\right)}^{2}}{2h}\right)$

$⇒ y = \lim_{h \to 0}\left(\dfrac{sinh \times 3 \times {\left(\dfrac{\pi}{2} - h\right)}^{2}}{2h}\right)$

We know that, $\lim_{h \to 0}\left(\dfrac{sinh}{h}\right) = 1$

$⇒ y = \lim_{h \to 0}\left(\dfrac{3 \times {\left(\dfrac{\pi}{2} - h\right)}^{2}}{2}\right)$

$⇒ y = \dfrac{3{\pi}^{2}}{8}$