Expand[(((((((E^(I t b))-(E^(I t a))))/(I t (b-a)))))^2)/.{a->((-1)/3),b->(1/3)}]

Expand[(((((((E^(I t b))-(E^(I t a))))/(I t (b-a)))))^3)/.{a->((-1)/3),b->(1/3)}]/(((-27)/(8 I)))//Expand

TrigReduce[ExpToTrig[(((((((E^(I t b))-(E^(I t a))))/(I t (b-a)))))^3)/.{a->((-1)/3),b->(1/3)}]]

Integrate[(1/2)(x-w),{x,-a,w}]//Simplify

Integrate[(1/2)(x-w),{x,-a,a}]//Simplify

(* sum the two the terms between ((-1)/3) and (1/3) *)
-(81/16)((((1/3)+w))^2)+(27/16)(((1+w))^2)//Simplify

(* sum the two the terms between (1/3) and 1 *)
-(81/12)w+(27/16)(((1+w))^2)//Simplify

(* sum the two terms greater than 1 *)
-(81/12)w+(27/4)w//Simplify