(a) Use complex integration techniques to evaluate the integral Sx dx x+ana > 0, 2, 3, 4,... Consider a sector contour of the form shown below, whose angle may depend on n. As part of your solution, consider that for n = 3, if 0=, then the integral on the straight arm of argument can be evaluated using the parametrization z = rei2/3,0<r< R, to be dz = مر ei2/3 dr r3+ a³ = -2/3 3 S² dx x³ + a³ y Reio reis Ө x R (b) Conditions on convergence of rational integrals: (i) Find the real number a such that => x² dx < ∞o (ii) If P(x) is a polynomial of order n = N, show that, as x → ∞, Pn(x) f(x) = → kx-m Pm(x) to put a condition on n and m for some constant k. (iii) Use your result from part so that f(x) dx doesn't diverge as the upper bound goes to infinity.