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5-2: Natural Logs, 2 Objectives: Assignment: 1. To integrate rational functions using the Log Rule • P. 338-340: 1-27 odd, 37, 43, 47-57 odd, 67, 87, 89, 91, 97-100 • P. 338-340: 29-35 odd, 2. To integrate trig functions 39, 46, 53, 69, 90 Warm Up Which of the following do not belong? 6𝑥 𝑑𝑥 3𝑥 2 + 1 6𝑥 3𝑥 2 6𝑥 3𝑥 2 + 1 2 𝑑𝑥 +1 𝑑𝑥 6𝑥 3𝑥 2 + 1 𝑑𝑥 Warm Up Which of the following do not belong? 6𝑥 𝑑𝑥 3𝑥 2 + 1 Cannot be integrated using a combination of 𝑢-substitution and the General Power Rule Objective 1 You will be able to integrate rational functions using the Log Rule Exercise 1 Find the derivative of 𝑦 = ln 2𝑥 + 3 . 𝑢>0 𝑦 = ln 2𝑥 + 3 2 𝑦′ = 2𝑥 + 3 𝑑 1 𝑑𝑢 𝑢′ ln 𝑢 = ∙ = 𝑑𝑥 𝑢 𝑑𝑥 𝑢 This is a rational function 𝑢′ 𝑢 Exercise 2 Find the antiderivative 2 𝑑𝑥 2𝑥 + 3 2 𝑑𝑥. 2𝑥+3 𝑥 ln 𝑥 = 1 1 𝑑𝑡 𝑡 ln 2𝑥 + 3 + 𝐶 𝑥>0 ln 2𝑥 + 3 + 𝐶 Log Rule for Integration Let 𝑢 be a differentiable function of 𝑥. 1 𝑑𝑥 = ln 𝑥 + 𝐶 𝑥 𝑢′ 𝑑𝑥 = ln 𝑢 + 𝐶 𝑢 1 𝑑𝑢 = ln 𝑢 + 𝐶 𝑢 Exercise 3 Find 2 𝑑𝑥. 𝑥 Exercise 4 Find 1 𝑑𝑥. 4𝑥−1 Exercise 5 Find the area the region bounded by the graph of 𝑥 𝑦 = 2 , the 𝑥 +1 𝑥-axis, and the line 𝑥 = 3. Exercise 6 Integrate the following. 1. 3𝑥 2 +1 𝑑𝑥 𝑥 3 +𝑥 2. sec2 𝑥 𝑑𝑥 tan 𝑥 Exercise 7 Integrate the following. 3. 𝑥+1 𝑑𝑥 𝑥 2 +2𝑥 4. 1 𝑑𝑥 3𝑥+2 Exercise 8 Which of the following does not belong? 2𝑥 𝑑𝑥 2 𝑥 +1 𝑥2 + 𝑥 + 1 𝑑𝑥 2 𝑥 +1 𝑥 𝑑𝑥 2 𝑥 +1 𝑥+1 𝑑𝑥 2 𝑥 + 2𝑥 Long Division If the degree of the numerator is greater than or equal to the degree of denominator, rewrite the integrand by dividing. 𝑥2 + 𝑥 + 1 𝑑𝑥 2 𝑥 +1 𝑥 1+ 2 𝑑𝑥 𝑥 +1 Exercise 9 Find 𝑥 2 +𝑥+1 𝑑𝑥. 2 𝑥 +1 Exercise 10 Find 2𝑥 𝑑𝑥. 2 𝑥+1 Exercise 11 Solve the differential equation 𝑑𝑦 𝑑𝑥 = 1 . 𝑥 ln 𝑥 Objective 2 You will be able to integrate trig functions Exercise 12 Find tan 𝑥 𝑑𝑥. Exercise 13 Find sec 𝑥 𝑑𝑥. Trig Rules sin 𝑢 𝑑𝑢 = − cos 𝑢 + 𝐶 cos 𝑢 𝑑𝑢 = sin 𝑢 + 𝐶 tan 𝑢 𝑑𝑢 = − ln cos 𝑢 + 𝐶 cot 𝑢 𝑑𝑢 = ln sin 𝑢 + 𝐶 sec 𝑢 𝑑𝑢 = ln sec 𝑢 + tan 𝑢 + 𝐶 csc 𝑢 𝑑𝑢 = − ln csc 𝑢 + cot 𝑢 + 𝐶 Exercise 14 Evaluate 𝜋/4 0 1 + tan2 𝑥 𝑑𝑥. Exercise 15 Find the average value of 𝑦 = tan 𝑥 on 𝜋 0, 4 . 5-2: Natural Logs, 2 Objectives: Assignment: 1. To integrate rational functions using the Log Rule • P. 338-340: 1-27 odd, 37, 43, 47-57 odd, 67, 87, 89, 91, 97-100 • P. 338-340: 29-35 odd, 2. To integrate trig functions 39, 46, 53, 69, 90