Need help with this homework problem for my differential geometry class. Please show work, thank you!

13. Consider the "spiral" a (t) = r(t) (cost, sint), where r is C and 0 co. b. Show that if r (t) = 1/(t + 1), then a has infinite length on [0, co). c. If r(t) = 1/(t + 1)2, does a have finite length on [0, co)? d. Characterize (in terms of the existence of improper integral(s)) the functions r for which a has finite length on [0, co). e. Use the result of part d to show that the result of part a holds even without the hypothesis that r be decreasing