lim x → 1 ∫ 0 ( x - 1 ) 2 t cos t 2 d t ( x - 1 ) sin ( x - 1 )

Applying L'Hospital's Rule, for numerator we have to apply Newton Leibnitz integral rule

$\underset{x\to 1}{\lim }\left(\frac{2(x-1)\times (x-1{)}^2\cos {\left(x-1\right)}^4-0}{(x-1)\cos (x-1)+\sin (x-1)}\right) \left(\frac{0}{0}\right)$

Divide $\left(x-1\right)$

$=\underset{x\to 1}{\lim }\frac{2(x-1{)}^2\cos (x-1{)}^4}{\cos (x-1)+{\displaystyle \frac{\sin (x-1)}{(x-1)}}}$

$=\frac{0}{1+1}$

$=0$