problem MARK BOX points HAND IN PART -3 653x5 5 NAME: Solutions 5 6 PIN: 7 % INSTRUCTIONS This exam comes in two parts. () HAND IN PART. Hand in only this part. () STATEMENT OF MULTIPLE CHOICE PROBLEMS. Do not hand in this part. You can take this part home to learn from and to check your answers once the solutions are posted. This exam has 3 types of problems: Problem, Multiple Choice, and Show All Your Work. On Problem, fill in the blanks. As you know, if you do not make at least half of the points on Problem, then your score for the entire exam will be whatever you made on Problem. For Multiple Choice problems, circle your answer(s) on the provided chart. No need to show work. The statement of multiple choice problems will not be collected. For the Show All Your Work problems, to receive credit you MUST: () work in a logical fashion, show all your work, indicate your reasoning; no credit will be given for an answer that just appears; such explanations help with partial credit () if a line/box is provided, then: show you work BELOW the line/box put your answer on/in the line/box (3) if no such line/box is provided, then box your answer. The mark box above indicates the problems along with their points. Check that your copy of the exam has all of the problems. Upon request, you will be given as much (blank) scratch paper as you need. During the exam, the use of unauthorized materials is prohibited. Unauthorized materials include: books, electronic devices, any device with which you can connect to the internet, and personal notes. Unauthorized materials (including cell phones, as well as your watch) must be in a secured (e.g. zipped up, snapped closed) bag placed completely under your desk or, if you did not bring such a bag, given to Prof. Girardi to hold for you during the exam (and they will be returned when you leave the exam). This means no electronic devices (such as cell phones) allowed in your pockets. At a student s request, I will project my watch upon the projector screen. Cheating is grounds for a F in the course. During this exam, do not leave your seat unless you have permission. If you have a question, raise your hand. When you finish: turn your exam over, put your pencil down and raise your hand. This exam covers (from Calculus by Thomas, 3 th ed., ET): 8.-8.5, 8.7, 8.8. Honor Code Statement I understand that it is the responsibility of every member of the Carolina community to uphold and maintain the University of South Carolina s Honor Code. As a Carolinian, I certify that I have neither given nor received unauthorized aid on this exam. I understand that if it is determined that I used any unauthorized assistance or otherwise violated the University s Honor Code then I will receive a failing grade for this course and be referred to the academic Dean and the Office of Academic Integrity for additional disciplinary actions. Furthermore, I have not only read but will also follow the instructions on the exam. Signature : Prof. Girardi Page of 5 Math
. Fill-in the blanks... arcsin( - ) -π 6 (Your answers should be an angle in RADIANS.).. Double-angle Formula. Your answer should involve trig functions of θ, and not of θ. sin(θ) sin θ cos θ.3. du u u ln u + C.. u n du n u n+ n+ +C.5. e u du e u +C.6. sec u du tan u + C.7. sec u du ln sec u + tan u or - ln sec u tan u + C.8. du a> a +u a ( ) tan- u a + C.9. Trig sub.: if the integrand involves u a, then one makes the substitution u a sec θ.. Integration by parts formula: u dv uv vdu Prof. Girardi Page of 5 Math
MULTIPLE CHOICE PROBLEMS Indicate (by circling) directly in the table below your solution to the multiple choice problems. You may choice up to answers for each multiple choice problem. The scoring is as follows. For a problem with precisely one answer marked and the answer is correct, 5 points. For a problem with precisely two answers marked, one of which is correct, points. All other cases, points. Fill in the number of solutions circled column. (Worth a total of point of extra credit.) Table for Your Muliple Choice Solutions Do Not Write Below problem a b c d e a b c d e 3 3a 3b 3c 3d 3e a b c d e number of solutions circled B x 5 5a 5b 5c 5d 5e 6 6a 6b 6c 6d 6e 7 7a 7b 7c 7d 7e 8 8a 8b 8c 8d 8e 9 9a 9b 9c 9d 9e a b c d e a b c d e a b c d e 3 3a 3b 3c 3d 3e 5 Extra Credit: Prof. Girardi Page 3 of 5 Math
. Evaluate ( + x) 3. Show your work below the box then put your answer in the box. ( ) 3 ( ) 5 ( ) + x + x + x + C 5 soln. Inspired by 8 Integrals number 3: ( + x) 8. Prof. Girardi Page of 5 Math
5. Show your work below the box then put your answer in the box. You may use part (a) in part (b). 5a. Evaluate sin x. sin x x sin x + C also ok x sin x cos x + C 5b. Evaluate x sin x. x sin x x x sin x cos x 8 + C also ok x x sin x cos x + sin x + C Is number 7 from the 8 Integrals. First Key Idea for Parts on the blue sheet. Prof. Girardi Page 5 of 5 Math
6. Part 6a should help with part 6b. Show your work below the box then put your answer in the box. 6a. Evaluate e (x ) (x). e (x ) (x) e x + C 6b. We cannot integrate the functions y e x and y x e x with techniques we have learned thus far (in fact, they do not have elementary antiderivatives). But we can integrate y (x + ) e x with techniques we know thus far. Evaluate (x + ) e x. (x + ) e x xe x + C Is number 8 from the 8 Integrals. Prof. Girardi Page 6 of 5 Math
STATEMENT OF MULTIPLE CHOICE PROBLEMS These sheets of paper are not collected. Hint. For a definite integral problems b f(x). a () First do the indefinite integral, say you get f(x) F (x) + C. () Next check if you did the indefininte integral correctly by using the Fundemental Theorem of Calculus (i.e. F (x) should be f(x)). (3) Once you are confident that your indefinite integral is correct, use the indefinite integral to find the definite integral. Hint. If a, b > and r R, then: ln b ln a ln ( ) b a and ln(a r ) r ln a.. Evaluate the integral soln. x x + 9. Prof. Girardi Page 7 of 5 Math
. Evaluate the integral soln. x x + 9. 3. Evaluate 3soln. π sin x cos 3 x. Prof. Girardi Page 8 of 5 Math
. Evaluate x x5 x 5 x AND specify the initial substitution. soln. The integrand has a u a, so we try u a sec θ. x 5 sec θ so 5 sec θ tan θ dθ Prof. Girardi Page 9 of 5 Math
5. Evaluate the integral π/ cos 3 x. 5soln. Break off one cos to form the du cos x and so u sin x. π/ cos 3 x [ sin x π/ 3 sin3 x [ sin π π 3 sin3 [ sin [ 3 sin3 [ 3 3 6. Evaluate the integral 8 x x +. Hint: complete the square in the denominator. 6soln. x x + (x ) 8 +. So 8. So let u x. x x+ (x) + 7. Evaluate the integral x ln x. 7soln. y ln x is easy to differentiate but hard to integrate so try parts with u ln x. Prof. Girardi Page of 5 Math
8. Evaluate the integral π/8 e 3x cos (x). 8soln. First we show that e 3x cos x e3x 5 (3 cos x + sin x) + C. We will use two integration by parts and the bring to the other side idea. For the two integration by parts, put the expontential function with either the u s both times or the dv s both times. Way # For this way, for each integration by parts, we let the u involve the expontenial function. u e 3x dv cos x du 3e 3x v sin x. So by integration by parts e 3x cos x e3x sin x 3 e 3x sin x. Now let to get u e 3x du 3e 3x dv sin x v cos x. e 3x cos x e3x sin x 3 [ e3x cos x 3 e 3x cos x e3x sin x + 3 e3x cos x 3 e 3x cos x. Now solving for e 3x cos x (use the bring to the other side idea) we get [ + 3 e 3x cos x e3x sin x + 3 e3x cos x + K and so Thus [ ( e 3x cos x 5 e3x sin x + 3 ) e3x cos x + K 5 e3x sin x + 3 [ K 5 e3x cos x + 5 [ e3x K ( sin x + 3 cos x) +. 5 5 e 3x cos x e3x 5 (3 cos x + sin x) + C. Prof. Girardi Page of 5 Math
Way # For this way, for each integration by parts, we let the dv involve the expontenial function. u cos x dv e 3x du sin x v 3 e3x. So, by integration by parts e 3x cos x 3 e3x cos x 3 e 3x sin x. Now let u sin x dv e 3x to get du cos x v 3 e3x. e 3x cos x 3 e3x cos x + [ 3 3 e3x sin x e 3x cos x 3 3 e3x cos x + 3 e3x sin x e 3x cos x. 3 Now solving for e 3x cos x (use the bring to the other side idea) we get [ + e 3x cos x 3 e3x cos x + 3 e3x sin x + K 3 and so [ ( e 3x 3 cos x 3 + 3 e3x cos x + ) 3 e3x sin x + K 3 5 e3x cos x + [ K3 5 e3x sin x + 3 + [ e3x K3 (3 cos x + sin x) + 5 3 + Thus e 3x cos x e3x 5 (3 cos x + sin x) + C. Doesn t Work Way If you try two integration by part with letting the exponential function be with the u one time and the dv the other time, then when you use the bring to the other side idea, you will get, which is true but not helpful. Prof. Girardi Page of 5 Math
Now evaluate the indefinite integral. So π/8 e 3x cos (x) [ e 3x 5 [ e 3π/8 (3 cos x + sin x) xπ/8 x 5 [ e 3π/8 5 () ( 3 cos π + sin π ) [ e 5 [ 5 (3) e3π/8 3 5 (3 cos + sin ) 9. Evaluate the integral 3/ 9soln. x ( x ) 3/. Evaluate the integral 3 3t + t + t 3 + t dt soln. Partial Fraction Decompostion. t 3 + t t (t + ) (t ) (t + ). So which gives 3t + t + t 3 + t A t + Bt + C t + A (t + ) + (Bt + C) t. t 3 + t 3t + t + A ( t + ) + (Bt + C) t, t : 3A + B t : C t : A and so B 3 3. Thus 3t + t + t + dt t 3 + t t dt + t + dt. Note 3 t + t + 3 t dt 3 [ dt t + dt + t + dt ln t + + tan t 3 [ ln + tan [ [ ln ln( 3 + tan ) + π [ π 3 3 ln. Prof. Girardi Page 3 of 5 Math
Also So 3 3t +t+ t 3 +t 3 3 dt dt lim t b + b t lim ln t b + 3 b lim dt diverges to infinity.. Let y p (x) be a polynomial of degree 5. What is the form of the partial fraction decomposition of b + [ ln 3 ln b. p (x) (x ) (x + )? Here A, B, C, D, E and F are constants. soln. First note degree of den. +() () 6 > 5 dgree of num. (i.e., we have strictly bigger bottoms) and so we do not have to first do long division. So next factor the dem.: (x ) (x + ) (x ) (x + ) (x + ) where x and x + are linear terms while x + is an irreducible quadratic term. Now see the partial fraction handout from class to see that the PDF takes the form A x + B x + + Cx + D x + + Ex + F (x + ).. Evaluate soln. x. diverges to. x This is an improper integral (the integrand y x defined at x ) and Note lim a + a x lim a + x a [ x [ + x x lim x a + a [ is continuous on (, ) (, ) but is not lim b b [ lim () a + [ + lim x a + a ( ) a b x. [ lim a + a. The graph of y is symmetric about the y axis and so lim x b ; or, you can x calculate this limit similarly to the limit we just calculated. So [ b [ x lim + lim [ + [. b x a + x 3. Evaluate 3soln. x x x 3. does not exist but also does not diverge to infinity. x 3 This is an improper integral (the integrand y is continuous on (, ) (, ) but is not x 3 defined at x ) and [ [ x [ b [ + lim + lim 3 x 3 x 3 b x 3 a + a x 3 a Prof. Girardi Page of 5 Math
Prof. Girardi Page 5 of 5 Math