Answer
$\cos 1\approx 0.54$
Work Step by Step
$\lim _{x\rightarrow 1^+}\dfrac {\int ^{x}_{1}\cos \theta d\theta }{x-1}=\dfrac {\dfrac {d}{dx}\left( \int ^{x}_{1}\cos \theta d\theta \right) }{\dfrac {d}{dx}\left( x-1\right) }=\dfrac {\cos x}{1}=\cos 1\approx 0.54$