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

${\left(\frac{1}{x}+x\sin x\right)}^{10}$

$\text{ Here, }n=10\text{ (even) }$

$\Rightarrow \text{ Middle term is }{\left(\frac{10}{2}+1\right)}^{th}=6^{\dot{h}}\text{ term }$

$T_6={}^{10}C_5{\left(\frac{1}{x}\right)}^5(x\sin x{)}^5$

$\Rightarrow 252(\sin x{)}^5=\frac{63}{8}\Rightarrow (\sin x{)}^5=\frac{1}{32}$

$\Rightarrow \sin x=\frac{1}{2}\Rightarrow \sin x=\sin \frac{\pi }{6}\Rightarrow x=n\pi +(-1{)}^n\frac{\pi }{6}$