$($c$)$ Using these choices for x$\_($k$)^(*)$ and $\backslash$Delta x$,$ the definition tells us that

$\backslash$int$\_1^52$xdx$=\backslash$lim$\_($n$->\backslash$infty $)[\backslash$sum$\_($k$=1)^$n f$($x$\_($k$)^(*))\backslash$Delta x$].$

What is f$($x$\_($k$)^(*))\backslash$Delta xk and n f$($x$\_($k$)^(*))\backslash$Delta x$=$

$($d$)$ Express $\backslash$sum$\_($k$=1)^$n f$($x$\_($k$)^(*))\backslash$Delta xn$.\backslash$sum$\_($k$=1)^$n f$($x$\_($k$)^(*))\backslash$Delta x$=$

$($e$)$ Finally, complete the problem by taking the limit as n$->\backslash$infty of the expression that you found in the previous part.

$\backslash$int$\_1^52$xdx$=\backslash$lim$\_($n$->\backslash$infty $)[\backslash$sum$\_($k$=1)^$n f$($x$\_($k$)^(*))\backslash$Delta x$]=$