(a) Give the mathematical definition of the statement "the function f : [0, +co) - R is locally Riemann integrable". (b) Let f : [0, too) - R be locally Riemann integrable. Give the mathematical definition of the statement "the function f is integrable over [0, too) in the improper sense". +00 Define the improper integral f(x) dx . Jo (c) Use the definition from (b) to prove the existence of the +00 improper integral dx and to evaluate this 1 + x2 integral. (d) State the Comparison Theorem for Improper Integrals. (e) State the theorem relating the integrability, in the improper sense, of the functions f and If|. (f) Prove the existence of the improper integral +0o COS X dx . 1 + x2 [Hint: you may find it helpful to use the results from (c), (d) and (e).] (g) Prove the existence of the improper integral .+00 - arctan x ) sin x dx. [Hint: you may find it helpful to integrate by parts.]