Find the coefficient of:
(i)  $x$ in the expansion of  $(1-2x^3+3x^5){\left(1+\frac{1}{x}\right)}^8$

Coeff of x in $(1–2x^3+3x^5)(1+\frac{1}{x}{)}^8$

= coeff of x in $(1+\frac{1}{x}{)}^8+$  coeff  of  x  in $(-2x^3)(1+\frac{1}{x}{)}^8+$ coeff of x in $3x^5(1+\frac{1}{x}{)}^8$

= coeff of x in $(1+\frac{1}{x}{)}^8–2$ coeff of $x^{-2}$  in  $(1+\frac{1}{x}{)}^8+3$ coeff of $x^{-4}$ in $(1+\frac{1}{x}{)}^8$

$r^{th}$ term in $(1+\frac{1}{x}{)}^8$

t_r = {^8C}_{r-1} ( \frac{1}{x})^{r-1} = {^8C_{r-1} x ^{1-r}

For  $1-r=1,r=0,$ coeff of $x=0$

For  $1-r=-2,r=3,$ coeff of $x={{}^{8}C}_2$

For  $1-r=-4,r=5,$ coeff of $x={{}^{8}C}_4$

Therefore  coeff of $x=0-2{{}^{8}C}_2+3\times {{}^{8}C}_4$
$=154$