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

$n = 10\left(even\right)$ 

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

$\begin{array}{l}{T}_6=7\frac{7}{8}\\ \Rightarrow T_{5+1}=\frac{63}{8}\\ \Rightarrow {10}_{C_5}{\left(\frac{1}{x}\right)}^{10-5}(x\sin x{)}^5=\frac{63}{8}\\ \Rightarrow 252\cdot \frac{1}{x^5}{x}^5\cdot {\sin }^{5}x=\frac{63}{8}\\ \Rightarrow 4{\sin }^{5}x=\frac{1}{8}\\ \Rightarrow {\sin }^{5}x=\frac{1}{2^5}\\ \Rightarrow \sin x=\frac{1}{2}=\sin \left(\frac{\pi }{6}\right)\\ \therefore x=n\pi +(-1{)}^n\frac{\pi }{6}\end{array}$