Each of these functions can be computed using the stack, or using algebraic entry, whichever way you prefer:
Computing ‘sin(x) / x’:
Using the stack: C-x ( <RET> S <TAB> / C-x ).
Using algebraic entry: C-x ( ' sin($)/$ <RET> C-x ).
Computing the logarithm:
Using the stack: C-x ( <TAB> B C-x )
Using algebraic entry: C-x ( ' log($,$$) <RET> C-x ).
Computing the vector of integers:
Using the stack: C-x ( 1 <RET> 1 C-u v x C-x ). (Recall that C-u v x takes the vector size, starting value, and increment from the stack.)
Alternatively: C-x ( ~ v x C-x ). (The ~ key pops a number from the stack and uses it as the prefix argument for the next command.)
Using algebraic entry: C-x ( ' index($) <RET> C-x ).