Let $f\left(x\right)=\left\{\begin{array}{l}\frac{\sin \left[x^2\right]\pi }{x^2-2x-18}+a{x}^3+b\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\text{ for }0\le x\le 1\\ 2\cos \pi x+{\tan }^{-1}x\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\text{ for }1<x\le 2\end{array}\right.$ is differentiable in $[0,2],\left[x\right]$ denotes the greatest integer less than or equal to $x\text{.}$ then

$f\left(x\right)=\left\{\begin{array}{l}a{x}^3+b\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\text{for }0\le x\le 1\\ 2\cos \pi x+{\tan }^{-1}x\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\text{for }1<x\le 2\end{array}\right.$

Since f(x) is continuous at $x=1\Rightarrow a+b=-2+\frac{\pi }{4}\to \left(1\right)$

Also f(x) is differentiable at $x=1\Rightarrow a=\frac{1}{6}$

$$\begin{gathered} From \left(1\right) \\ \frac{1}{6}+b=-2+\frac{\pi }{4} \\ \Rightarrow b=-2-\frac{1}{6}+\frac{\pi }{4} \\ \Rightarrow b=\frac{\pi }{4}-\frac{13}{6} \end{gathered}$$