(Solved): This part uses contour integration to calculate 0 x sin x 1 + x2 -dx. - Consider the integral zeiz ...

This part uses contour integration to calculate 0 x sin x 1 + x2 -dx. - Consider the integral zeiz YR +22dz, where yn is the contour formed by the straight line joining (-R, 0) and (R,0), together with the half circumference |z| = R, Im (2) > 0, oriented counter- clockwise. (i) Show that the integral over the half circumference in VR goes to zero as R goes to infinity (ii) Use the deformation of contours Theorem to replace VR by a suitable cir- cumference. (iii) Use Cauchy's Theorem to calculate the integral over the new contour (cir- cumference). (You can also work it out directly if you prefer.) (iv) Use (i), (ii) and (iii) to evaluate x sin x L 1 + x2dx.