For each a, b E R, consider B(a, b) = f t-1 (1 t)b-1 dt. (a) For which values of a and b the integral B(a, b) is an improper integral? (b) Show that the integral converges when a > 0 and b > 0. (c) Show that the integral diverges if either a 0 or b 0. (d) Using t= sin(0)2 show that for a, b> 0 we have /2 255/2 B(a, b) = 2 sin (0)2a-1 cos(0)26-1 do.