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

$\begin{array}{l}{\left(\frac{1}{x}+x\sin x\right)}^{10}\mathrm{n}=10\text{ is even }\\ \text{ M. }\mathrm{T}=T_{\frac{10}{2}+1}=T_{5+1}=7\frac{7}{8}\\ \Rightarrow 10c_5{\left(\frac{1}{x}\right)}^5(x\sin x{)}^5=\frac{63}{8}\\ \Rightarrow \frac{10\times 9\times 8\times 7\times 6}{5\times 4\times 3\times 2\times 1}\frac{1}{x^5}{x}^5{\sin }^{5}x=\frac{63}{8}\\ \Rightarrow 63\times 4{\sin }^{5}x=\frac{63}{8}\end{array}$

$\begin{array}{l}\Rightarrow {\sin }^{5}x=\frac{1}{32}\\ \Rightarrow \sin x=\frac{1}{2}\Rightarrow x=n\pi +(-1{)}^n\frac{\pi }{6},n\in z\end{array}$