$\text{ If the middle term of }{\left(\frac{1}{x}+x\sin x\right)}^{10}\text{ is equal to }7\frac{7}{8},\text{ then the value of }x\text{ is }$

n = 10(even)

$\text{ Middle term }=\frac{10}{2}+1=5+1=6$

$T_6=7\frac{7}{8}$

$\Rightarrow T_{5+1}=\frac{63}{8}$

$\Rightarrow {10}_{c_5}{\left(\frac{1}{x}\right)}^{10-5}(x\sin x{)}^5=\frac{63}{8}$

$\Rightarrow 252.\frac{1}{x^5}{x}^5.{\sin }^{5}x=\frac{63}{8}$

$\Rightarrow 4{\sin }^{5}x=\frac{1}{8}$

$\Rightarrow \left(\sin x\right)=\frac{1}{2}$

$\therefore x=n\pi +{(-1)}^n\frac{\pi }{6}$,n$\in Z$