$\lim_{x \rightarrow \frac{\pi}{2}}\frac{\int_{\pi/2}^{x}{(2^{- \cos t}{- 1)\ dt}}}{\int_{\pi^{2}/4}^{x^{2}}{\left( \sqrt{t} - \frac{\pi}{2} \right)dt}}$ is equal to

Question

Dr. Abubacker · Accepted Answer

(b)
$\lim_{x \rightarrow \frac{\pi}{2}}{\frac{\int_{\pi/2}^{x}{(2^{- \cos t}{- 1)dt}}}{\int_{\pi^{2}/4}^{x^{2}}{\left( \sqrt{t} - \frac{\pi}{2} \right)d}t} = \lim_{x \rightarrow \frac{\pi}{2}}\left( \frac{2^{- \cos x}{- 1}}{\left( x - \frac{\pi}{2} \right) \bullet 2x} \right)}$
Applying L’ Hospital’s rule
$= \lim_{n \rightarrow \frac{\pi}{2}}\frac{2^{- \cos x}(\log_{e}{2)(\sin{x)}}}{2x + 2\left( x - \frac{\pi}{2} \right)} = \frac{\log_{e}2}{\pi}$

Q. $\lim_{x \rightarrow \frac{\pi}{2}}\frac{\int_{\pi/2}^{x}{(2^{- \cos t}{- 1)\ dt}}}{\int_{\pi^{2}/4}^{x^{2}}{\left( \sqrt{t} - \frac{\pi}{2} \right)dt}}$ is equal to

Explanation

Reviewed by Dr. Abubacker

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