a. Consider the indefinite integral int(sinx*e^(cosx))dx. It is = -int(e^(cosx))d(cosx) = -e^(cosx) (+C). So integral in question is a limit as A->+infinity of -e^(cosA) + e^(cos0) = e - e^(cosA). This limit doen't exist as cos(A) takes any values between -1 and 1 for arbitrary large A. Answer: diverges.
b. Here a critical point is x=1 where lnx=0 stays at denominator. Find the indefinite integral first: int(1/(x*lnx))dx = int(1/lnx)d(lnx) = int(1/u)du = lnu (u = lnx). So it is ln(ln(x)). The definite integral is the limit of ln(ln(2)) - ln(ln(a)) when a tends to 1+0 (to 1 from the right). But ln(a) tends to +0 as a tends to 1+0, so ln(ln(a)) tends to -infinity. Answer: diverges. (or we can say equals +infinity, as the limit not finite but exists)