Your post is a bit muddled but all you need to do is work out
which you already had correct and it does give the answer you quoted.
(-1)^6 = 1 and -2 x -1 =2
so you have 1/6+2= 2 1/6
By the way you normally sub in the upper limit first so in this case the area is negative (below the x axis) but the answer you should give is the absolute value as given in the book 2 1/6.