[ _x 2 ( 1 ( x - 2 )^2_x^3^( 2 )^3 ( t^1 3 )dt )] is equal to

Question

$$\mathop {\lim }\limits_{x \to {\pi  \over 2}} \left( {{1 \over {{{\left( {x - {\pi  \over 2}} \right)}^2}}}\int\limits_{{x^3}}^{{{\left( {{\pi  \over 2}} \right)}^3}} {\cos \left( {{t^{{1 \over 3}}}} \right)dt} } \right)$$ is equal to

Accepted Answer

Using L'hospital rule

$$\begin{aligned}
& =\lim _\limits{x \rightarrow \frac{\pi^{-}}{2}} \frac{0-\cos x \times 3 x^2}{2\left(x-\frac{\pi}{2}\right)} \
& =\lim _\limits{x \rightarrow \frac{\pi^{-}}{2}} \frac{\sin \left(x-\frac{\pi}{2}\right)}{2\left(x-\frac{\pi}{2}\right)} \times \frac{3 \pi^2}{4} \
& =\frac{3 \pi^2}{8}
\end{aligned}$$

Solutions · 1