Please write clearly and explain all the steps! Thank you!

6. (20 points) Consider the (real-variable) integral 1 : = Vx(1 -x) dx, where the integrand takes positive values by definition. a. Relate I to the contour integral J := Vz(z - 1) dz 2 =0 2 = C over the contour C shown at the right, where the integrand is understood to behave like Vz(z - 1) ~ z in the asymptotic limit z - co. Hint: Argue that V(a tic) (x tie - 1) = tivx(1 -x) +O(6) when z = x tie is just above or below the branch line in the diagram above, i.e., with 0 0. b. Deform the contour C to a large circle CR := { z c C (z| = R }, and use the residue theorem to evaluate J. Conclude that I = 7 /8. c. Repeat this argument to show that, for any integer n 2 0, In : = x" Vx(1 - x) dx = -1 bn, where bn denotes the coefficient of en+2 in the Taylor series expansion of V1 - E