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3. (a) (i) Determine 1 1= _It a212 dx in terms of the positive constant a. (ii) Hence, show that I = 1 when f(a) = 2 arctan(a?) - a = 0. (iii) Use Newton's method (with an initial guess of do = 0.3) to determine the first positive solution of f(a) = 0, correct to four decimal places. (b) A tank initially contains 50 litres of pure water. Salt water, containing a constant k kilograms of salt per litre, is pumped in at a rate of 2 litres per minute. A salty mixture flows out at the same rate of 2 litres per minute. The amount A of salt (in kilograms) in the tank at time t minutes satisfies the first order differential equation dA dt + 25 A = 2k with initial condition A(0) = 0. (i) Determine the amount of salt in the tank at any time t. (ii) Find the amount of salt in the tank in the limit as too