Math 1B Discussion Exercises GSI Theo Johnson Freyd http math berkeley edu theojf 09Spring1B Find two or three classmates and a few feet of chalkboard Introduce yourself to your new friends and write all of your names at the top of the chalkboard As a group try your hand at the following exercises Be sure to discuss how to solve the exercises how you get the solution is much more important than whether you get the solution If as a group you agree that you all understand a certain type of exercise move on to later problems You are not expected to solve all the exercises some are very hard Many of the exercises are from Single Variable Calculus Early Transcendentals for UC Berkeley by James Stewart these are marked with an Others are my own or are independently marked Always draw pictures Integration by Parts In 1A you learned the product rule for differentiation d d d f x g x f x g x f x g x dx dx dx This is often abbreviated f g 0 f 0 g g 0 f or d f g f dg g df By the fundamental theorem of of a function is the original function Thus f g C R calculus R the integral of Rthe derivative R d f g f dg g df f dg g df Rearranging the equality gives the integration by parts formula which is a kind of product rule for indefinite integrals Z f x g 0 x dx f x g x What about definite integrals Then Z b 0 Rb a f dg f x g x dx a Rb a g df f x g x ba Z g x f 0 x dx Rb d f g f g ba f b g b f a g a a Z b g x f 0 x dx a 1 a Use the integration by parts formula to integrate Z x ex dx Hint let f x x and g 0 x ex What is g x R R b RBased on your answer to part a find x2 ex dx Based on that find x3 ex dx and x4 ex dx c Guess R n xthe pattern from R part b Prove your pattern use integration by parts to write x e dx in terms of xn 1 ex dx This is an example of a reduction formula 2 a Use the integration by parts formula to integrate Z ln x dx Hint let f x ln x and g 0 x 1 1 b Use the integration by parts formula to integrate Z ln x 2 dx Hint let f x ln x 2 and g 0 x 1 c Use the integration by parts formula to integrate Z ln x n dx R 3 Find a formula for f x g 00 x dx by applying the integration by parts formula twice Sup0 0 00 pose that f 1 R 4 2 f00 4 7 f 1 5 and f 4 3 and that f x is continuous on 1 4 What is 1 x f x dx R x R x 4 By integrating by parts twice and rearranging find e cos x dx What is e sin x dx How R ax R x about e cos x dx where a is a constant How about x e cos x dx 5 Use integration by parts to evaluate the following integrals For some you will first need to make a substitution Z Z Z a t sin 2t dt b x2 sin x dx c arcsin x dx Z Z 1 Z 9 ln y s 2 x d s 2 ds e x 1 e dx f dy y 0 4 Z 3 Z Z t x3 cos x dx h g arctan 1 x dx i es sin t s ds 0 Z j 1 3 cos 2 d 2 Z k 0 ecos t sin 2t dt Z l sin ln x dx 0 6 What s wrong with the following proof that 0 1 Z Z Z 1 1 1 1 dx x x dx 1 dx 1 ln x ln x 2 x x x x 7 Why can we forget to addR an arbitrary constant during the intermediate steps when integrating by parts Integrate xn ex dx completely honestly let u xn and dv ex dx but this time let v ex C R 2 8 Let n 2k 1 be an odd integer Calculate x 0 cosn x dx in two different ways R a UsingR a reduction formula What happens to the boundary terms the uv in u dv uv v du Why does it matter that n is odd The formula is different for even n b Using a u substitution Hint cos2 x 1 sin2 x and sin0 x cos x For any given k you could then expand out and evaluate the integral For a general k you can use integration by parts to get a reduction formula R 2 R 2 9 What is 0 dx Use the reduction formula from the previous problem to evaluate 0 cosn x dx for n even 2
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