Integration Problems and Substitutions for MAT 168 Exam, Exams of Analytical Geometry and Calculus

Download Integration Problems and Substitutions for MAT 168 Exam and more Exams Analytical Geometry and Calculus in PDF only on Docsity! Unit 2 Test MAT 168 Name: Instructions: No calculators or notes are allowed on the test, and all work must be the student’s own. Leave all answers exact. Write out any substitutions or identities used. Remember that in many circumstances, you should be able to doublecheck your answers. In order to receive full credit for each problem of the exam, you must: a. Show legible and logical (relevant) justification which supports your final answer. b. Use complete and correct mathematical notation Problem Points Points Possible 1 16 2 24 3 30 4 32 Total 100 Problem 1 (14 points) Circle the correct answer. On this problem, you do not need to show any supporting work. Consider ∫ π/3 0 cosx √ sinx dx. Which of the following is an equivalent integral? a. ∫ 1/2 0 √ u du b. ∫ π/3 0 √ u du c. ∫√3/2 0 √ u du d. ∫ 1/2 0 u du If we use the substitution x = 3 sin θ to evaluate ∫ √ 9−x2 x2 dx, which is the resulting integral? a. ∫ cot2 θ dθ b. 13 ∫ cot θ csc θ dθ c. ∫ cot θ dθ d. 3 ∫ cot θ cos θ dθ Suppose we make the trigonometric substitution x = 2 tan θ to evaluate ∫ 1 x2 √ x2+4 dx. If the answer we get out is − csc θ4 + C, how will this answer look when expressed in terms of x instead of θ? a. − csc x4 + C b. − x√ x2+4 + C c. − √ x2+4 4x + C d. tan −1 (x 2 ) + C Suppose that in attempting to evaluate ∫ sin x x dx using integration by parts, we let u = sinx and dv = 1 x dx. The resulting first step would be: a. − sin xx2 − ∫ cos x x2 dx b. ln |x| sinx− ∫ ln |x| cosx dx c. ln |x| sinx− ∫ cos x x dx d. − cos x x − ∫ cos x x2 dx Problem 2 (24 points) Determine the following integrals. a. ∫ sin2 x dx b. ∫ x(x2 − 2)5 dx c. ∫ sin3 x cos3 x dx d. ∫ ex ex + 1 dx