please help with this integral questions, it is too difficult>_

k 2. (24 Points) For any non-negative integer k, let fk(x) = V1 - and define Gk : (-1, 1) - R by Gk(x) = fx(t) dt. (a) (6 points) Using integration-by-parts, or otherwise, show that for any k 2 2, GK(x) = - 1 k-1 1-12 + k - 1 k "GK-2(x ) . k (b) (6 points) Show that lim Gk(x) exists for each k. In addition, compute I+1- lim Gs(x). I-1- (c) (12 points) Given that f2(x) = T(2-12) 2+ 12 3x (4+ 12) (1 -22) 3/2' $2 (2) = (1 - 12) 5/2' $2 () = (1-12)7/2 (i) Determine the interval(s) of strictly increase, and the interval(s) of strictly decrease of G. Hence determine the local extrema of G, if any. (ii) Determine the interval(s) where G is convex, and the interval(s) where it is concave. Hence determine where G has an inflection point, if any. (iii) Find the second Taylor polynomial of G2 with reference point 0, and use it to estimate G2(-0.2). In addition, find an upper bound for the absolute error committed