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  3. 2 we learned in class how to integrate any rational...

Question: 2 we learned in class how to integrate any rational...

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2. We learned in class how to integrate any rational function using partial fractions. We also noted that any integrand which can be converted to a rational function is then also integrable. Here we will introduce the Weierstrass substitution, t tan ( that converts some trigonometric integrals to rational functions. (o) By writing co show that 1 + t2 and also that 2t sin r- Hint: Consider the quantities cos 20 and sin 26 (b) We will now employ the results of part (a) to integrate dr -1/v21+2 i. Using the trigonometric substitution sin u, show that T/4 1 + cos 2u -/4 3cos 2u

3. In an accident, a chemical plant releases an amount of pollutant A into a canal at time on the canal a distance aro downstream is given by v Akt exp (-a ) E(t) C(t) = exp (-R t where k is a positive constant. (a) Calculate li m C() and lim C(t) and explain the resnits in relation to the amount of pollutant in the water at time t and location ro (b) The town council decides to evacuate the town before the pollutant reaches its maximum level at the town. How long (from the time of the pollutant release) does the council have to evacuate the town? (c) Sketch a graph of C(t) for t0. [You can set k 1 here for simplicityl

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