# Pts Evaluate the following integrals: a Z tan 2 x sec 2 x...

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Math 1910, Final Aug 05, 2013, 8:30 AM to 11:00AM For maximum credit, show all the steps leading to the final answer and write clearly and legibly. 1. (30 pts) Evaluate the following integrals: (a) Z tan 2 x sec 2 x dx (b) Z cos x sin x + sin 2 x dx (c) Z arctan x dx (d) Z 1 - 1 sin(5 πx ) cos(2 πx ) dx (e) Z e x cos x dx 2. (15 pts) Find the sum of the following series (a) X n =0 Z n +1 n dx 1 + x 2 (b) 1 + e 2 + e 4 2! + e 6 3! + e 8 4! + e 10 5! ... (c) 1 + e + e 2 + e 3 + e 4 ... 3. (15 pts) (a) Write the first two terms of the Taylor series of ln x about the center x = 1. This is the approximation of the function around x = 1 by a first order polynomial. (b) Write the equation of the line tangent to the function y = ln x at x = 1. (c) Use part(a) to find the approximate value of ln(1 . 1). What can you do to find a better approximation to ln(1