University of California Los Angeles Fall 2025 Instructor Dr Johnson Date Oct 24 2025 Name UCLA ID Signature MATH 31B Integration and Infinite Series Midterm 1 This exam contains 10 pages including this cover page and 3 problems Check to see if any pages are missing Enter all requested information on the top of this page and also Put your initials on the top of every page in case the pages become separated This is a closed book exam except for one 3in by 5in index card lled in with your own handwriting There is also a provided formula sheet No calculators phones laptops smartwatches etc are allowed Problem Points Score 1 2 3 Total 23 17 40 80 The following rules apply Unless the problem explicitly states otherwise you must show your work to justify your answers Mysterious or unsupported answers will not re ceive full credit An incorrect answer supported by substantially correct calculations and explana tions might still receive partial credit Organize your work in a reasonably neat and coherent way in the space provided Work scat tered all over the page without a clear ordering will receive very little credit If you need more space you may use the blank pages at the end of the exam clearly indicate when you have done this Good luck MATH 31B Oct 24 2025 Midterm 1 Page 2 of 10 Initials Integrals cid 90 cid 90 cid 90 cid 90 cid 90 cid 90 cid 90 cid 90 cid 90 cid 90 dx 1 x2 dx x2 1 sin 1 x C tan 1 x C x 2 x 2 1 2 1 2 sin2 x dx sin x cos x C sin x cos x C cos2 x dx tan x dx ln sec x C cot x dx ln sin x C sec x dx ln sec x tan x C csc x dx ln csc x cot x C sec2 x dx tan x C csc2 x dx cot x C MATH 31B Oct 24 2025 Midterm 1 Page 3 of 10 Initials 1 a 5 points Mark ALL the correct logarithm and exponent properties cid 13 log5 ab b log5 a cid 13 log5 a b log5 a log5 b cid 13 log5 ab log5 a log5 b cid 13 5x5y 5xy cid 13 5x y 5xy b 3 points What is lim x 0 ln x c 5 points What is the derivative of f x ex3 5x Answer for b d 5 points Evaluate Answer for c cid 90 1 5 1 x dx Answer for d cid 90 dx x x 5 2 x2 8 e 5 points Brenda wants to integrate using partial fraction integra tion Write down the form of the decomposition she should use You do not need to solve for the coe cients Answer for e MATH 31B Oct 24 2025 Midterm 1 Page 4 of 10 Initials 2 17 points Compute the following limit Ensure you clearly label if when you use L H opital s Rule 1 sin 4x 1 x lim x 0 answer MATH 31B Oct 24 2025 Midterm 1 Page 5 of 10 Initials 3 Evaluate the following inde nite integrals a 15 points cid 82 ln x cos ln x x dx Answer for a MATH 31B Oct 24 2025 Midterm 1 Page 6 of 10 Initials cid 82 b 10 points 5 25 4x2 dx Answer for b MATH 31B Oct 24 2025 Midterm 1 Page 7 of 10 Initials cid 82 c 15 points 1 t2 4 t2 dt Answer for c MATH 31B Oct 24 2025 Midterm 1 Page 8 of 10 Initials This page intentionally left blank MATH 31B Oct 24 2025 Midterm 1 Page 9 of 10 Initials This page intentionally left blank MATH 31B Oct 24 2025 Midterm 1 Page 10 of 10 Initials This page intentionally left blank