Lab 12: Substitution Rule Show all work on a separate sheet of paper with justifications where necessary. Answers should be clearly marked or boxed. Print first and last names of each group member at the top of the page. 1. Evaluate the integral S (sin x cos x) dx two different ways. First, let u = sin x, then let u = cos x. Show that the apparently different answers are equivalent. 2. A bottle of wine at room temperature (70 F) is placed in a refrigerator at 4 pm. Its temperature after t hours is changing at the rate of -18e-0.65t degrees Fahrenheit per hour. (a) By how many degrees will the temperature have dropped by 7 pm? (b) What will the temperature of the wine be at 7 pm? (c) Let n(t) = -18e-0.65t and find its antiderivative, N(t). (d) Graph n(t) and N(t), including the horizontal asymptotes. (e) Given the context of this problem, why does it make sense that these horizontal asymptotes exist? What is happening to the temperature as t - co? What is happening to the rate of change of the temperature as t - co? (f) Where on each of the graphs n(t) and N(t) can the solution to part (a) be found? Point them out. 3. Evaluate the definite integral Sa (cx + da) dx, with a, b, c, a constant, two different ways. First, use integration by substitution, then find another way to integrate and confirm your solution. Lab 12 Supplement: Substitution Rule 1. Find the antiderivative of f(x) = sec x tan x three different ways. That is, use three different u-substitutions. Namely, u = tanx, u = secx, and u = secx. One of the answers may initially look different from the other two, but explain why it is equivalent. 2. Evaluate the definite integral So (xvx + 3) dx two different ways, using two different u-substitutions. Namely, u = x + 3 and u = vx + 3. Hint for using u = vx + 3: Multiply the integrand by a fraction that equals 1