Math 1B Worksheet 4 Integration by Parts Thursday 6 September 2007 Theo Johnson Freyd theojf math berkeley edu Please introduce yourselves to each other and put your names at the top of a piece of blackboard Take turns being the scribe each of you should have a chance to write on the chalkboard for at least one of the exercises These exercises are hard harder than on the homework quizzes or exams That means that you should spend some time thinking and talking about them they re designed to be solved in groups the best way to learn mathematics The problems are roughly in order of increasing difficulty I don t expect any group to solve all of them Don t forget to draw pictures 1 Integrate Z dx 3 cos x 4 sin x 2 Let s find the partial fraction decomposition of say x3 x 1 x2 2x a Factor the denominator to find a b and c x 1 x 1 x3 x2 2x x a x b x c b We want to write x3 x 1 A B C 2 x 2x x a x b x c We 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 you about 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 about x3 3 Solve x 1 2x2 x Z x3 x 1 x2 x x dx x2 1 in 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 integrate dv to find v with whatever method you want 4 When a marble with mass m say falls through a viscous liquid like honey a constant downward force gravity minus buoyancy g mg b acts on it and friction impedes its motion with a force proportional to the square of the marble s velocity say v 2 Then the marble s velocity is given by the equation Z Z m dv dt g v 2 To simplify the problem let s let m g 1 or if you want to do the problem with all the unknown constants Assume that the marble starts at rest at time t 0 Solve this integral to find v as a function of t don t forget to use the fact that v 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 integrate Z dx x2 1 by a trig substitution Try factoring the denominator with complex numbers and using integration by parts 2
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