Math 1B Worksheet 4:Integration by PartsThursday, 6 September 2007Theo [email protected] introduce yourselves to each other, and put your names at the top of a piece ofblackboard. Take turns being the scribe: each of you should have a chance to write on thechalkboard for at least one of the exercises.These exercises are hard — harder than on the homework, quizzes, or exams. Thatmeans that you should spend some time thinking and talking about them; they’re designedto be solved in groups (the best way to learn mathematics). The problems are roughly inorder of increasing difficulty. I don’t expect any group to solve all of them.Don’t forget to draw pictures.1. Integrate:Zdx3 cos x + 4 sin x2. Let’s find the partial fraction decomposition of, say,x + 1x3+ x2− 2x(a) Factor the denominator to find a, b, and c:x + 1x3+ x2− 2x=x + 1(x − a)(x − b)(x − c)(b) We want to writex + 1x3+ x2− 2x=Ax − a+Bx − b+Cx − cWe could put everything over a common denominator (i.e. multiply by (x3+x2− 2x) and compare like terms. Instead, multiply both sides by just (x − a),and then plug in x = a into both sides of the equation. What does this tell youabout the values of A, B, and C?1(c) Use this method to find A, B, and C.(d) How would you do this for other examples? What aboutx + 1x3− 2x2+ x?x + 1x3− x2+ x?3. SolveZx dxx2− 1in three different ways:(a) With a u-substitution.(b) With a trig substitution.(c) By decomposing the integrand into partial fractions.For an extra challenge, try doing it by parts: u = x, dv = dx/(x2− 1), and integratedv to find v with whatever method you want.4. When a marble (with mass m, say) falls through a viscous liquid like honey, a constantdownward force (gravity minus buoyancy = ˜g = mg − b) acts on it, and frictionimpedes its motion with a force proportional to the square of the marble’s velocity(say αv2). Then the marble’s velocity is given by the equationZdt =Zm dv˜g − αv2To simplify the problem, let’s let m = ˜g = α = 1 (or, if you want to, do the problemwith all the unknown constants). Assume that the marble starts at rest at timet = 0. Solve this integral to find v as a function of t; don’t forget to use the fact thatv(t = 0) = 0 to figure out the constant of integration.For an extra challenge, find the distance the marble has fallen as a function of t.5. This problem is only for those who know complex numbers.Normally we integrateZdxx2+ 1by a trig substitution. Try factoring the denominator with complex numbers andusing integration by
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