This problem is concerned with evaluating some improper integrals. In particular you will use an improper integral over an interval of infinite length to evaluate an integral of a function not defined at one end point. This will involve a special function I which arises in many applications in the sciences. a. Evaluate f(log x) dx. b. Explain how you would evaluate fr8 (log x) dx, but do not actually compute it. Would your method work if the exponent '8' were replaced by '8', or if '7' were replaced by '7'? Why? c. Define the function F(t) = f e-t- dx for t> 0. i. Calculate I(1). ii. Lett be a fixed number, 0 < t < 1. Explain why I (t) is an improper integral. Then, show that the integral that defines I'(t) is convergent. iii. For t > 0, show that I'(t + 1) = tr(t). d. Let n be a positive integer. Use part (c(iii)) to find an equation relating I(n) and n!. Read about mathematical induction and prove your equation is correct. e. Let m be a positive real number and let n be a positive integer. Find an equation relating (log x)" da to I and use this to find a formula for this integral in terms of m and n. f. Use part (e) to evaluate three integrals in (a) and (b).