lim (∫ (sin(2t^{1/3} + cos (t^1 / 3)) dt ; x ∈[ x^3, (π / 2^3)] / (x - π / 2)^2) is equal to :

Question

\(\lim\limits _{x \rightarrow \frac{\pi}{2}}\left(\frac{\int_{x^{3}}^{(\pi / 2)^{3}}\left(\sin \left(2 t^{1 / 3}\right)+\cos \left(t^{1 / 3}\right)\right) \mathrm{dt}}{\left(x-\frac{\pi}{2}\right)^{2}}\right)\) is equal

to :

(1) \(\frac{9 \pi^{2}}{8}\)

(2) \(\frac{11 \pi^{2}}{10}\)

(3) \(\frac{3 \pi^{2}}{2}\)

(4) \(\frac{5 \pi^{2}}{9}\)  

Answer 1

Correct option is : (1) \(\frac{9 \pi^{2}}{8}\) 

\(\lim \limits_{x \rightarrow \frac{\pi}{2}} \frac{0-\{\sin (2 x)+\cos (x)\} \cdot 3 x^{2}}{2\left(x-\frac{\pi}{2}\right)}\)

\(=\lim \limits_{x \rightarrow \frac{\pi}{2}} \frac{-\{2 \sin x \cos x+\cos x\} 3 x^{2}}{2\left(x-\frac{\pi}{2}\right)}\)

\(=\lim\limits _{x \rightarrow \frac{\pi}{2}}\left\{\frac{2 \sin x \sin \left(\frac{\pi}{2}-x\right)}{2\left(x-\frac{\pi}{2}\right)}+\frac{\sin \left(\frac{\pi}{2}-x\right)}{2\left(\frac{\pi}{2}-x\right)}\right\} 3 x^{2}\)

\(=\left(1(1)+\frac{1}{2}\right) 3\left(\frac{\pi}{2}\right)^{2}\)

\(=\frac{9 \pi^{2}}{8}\)