# Marks letting u tan x evaluate z tan x sec 2 xdx

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2. [6 marks ] Letting u = tan x, evaluate Z tan x sec 2 xdx using:
(a) The substitution rule. (b) Integration by parts.
3. [7 marks ] One model for the spread of a rumour is given by dt dx = - k x ( x - N ) , where x is the number of people who have heard the rumour, N is the population size, and t is time (in hours). Suppose N = 256 , the total number of students currently registered in MA103, and k = 160 . (a) Find the partial fraction decomposition of dt dx . [Integration is not required.]
(b) Suppose at time t = 0 , one MA103 student decides to start a rumour that the final exam will be open-book [this is just a rumour!]. Using this information and the result from part (a), find time t as a function of x. [Note: Don’t forget the absolute value bars in your result!]