UCR - MATH 009B - Study Guide - Final

View Full Material##### School: University of California Riverside

##### Department: Applied Mathematics

##### Course: First-Year Calculus

##### Professor: Thunwa Theerakarn

##### Term: Fall 2018

##### Tags: Math and Calculus

##### Name: Math 9B Final Study Guide

##### Description: These notes go over sections from chapter 6 and chapter 7 that were covered after the midterm.

##### Uploaded: 12/05/2018

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Unformatted text preview: 64 Math 96 Final Inigonometric Substitution then dx = a coscende - I Pyou see vast? use substitution X-a-Since) Land Vas-y? = a.casco) then dxea sec?(Q)do FIf you see vara? use substitution x=a tance) and X2+a? = a.secle) then dx= a secletan () do If you see vaba? use substitution X-a secco Land va? = a tan (A) Jhome dx = ? First, figure out which substituhon to use. V 57x? fits the form Jx?ta? closest Since a?=5 then a=15 Find out what a is to use proper substitution SO X=5 tan (0) dx= 15. sec (0) do and X-5. tan? (0) Nowy plug in substitutions and solve 15+5.tan l0) 2 sec @de dx sec? 15. sec?(edo J15. J + tano 5 sec Code J18. secie) - J seccosdo - in sec (0) Fan (0) + Crtan(A)= x 20 = ln 5+ x + x + 5 7 4 sec(0)=15+X? 2*+ 156.5 Partial Fraction Decomposition (x-a)is its highest power, then -Linear terms. If xa divides the denominator, and . Al, Az An x-a Gra]? Gaan -Quadratic terms: If x2+bx+c divides the Bix + C denominatag and (?+bx+c)" is its highest power, then Bzx+C2 on Box +Cn 1x2+bx+c * 62+bx+c) (x2 +bx+c)"| Ma. Find partial fraction decomposition of XP-1 (x+1/x-D L A VB Need to find (x+1)(x-1) 4 x+ and B hest multiply by (x+1)(x-1) fotoext), quxft). Atatt) (x1). B fta- tt 1 = Alx+1)+ Bx-) = Ax+A + Bx-B. Ox + 1 = (A + B)x + (A-B) Set equal coefficients of each power of x on both sides O {A+B=0A--B subin -B-B---23=1 B=-1/2 OLA-BELA--Bar-f*)= -> A= 30 T V2 -V2 (x+1)(x-D *- 7 x + *b. Vyz7 dx = f(x)Q-1)dx=2 y dx - Juridx u-x-T =Y/2.8 Yu du-42 pwdu du-dx = 1/2 In lul-Y2Inlwl+C x+1=Y2 In 1x-1|-22 in 1x +1] + C) ffinat answer for integral WE16.8 Improper Integrals Flet Hou be a combinuous function from any a to b +-7-00 to f@dx = limo fodx PG)dx= f(x) dx + pop foodx -If the integral has a hnite values converges. Otherwise, it is divergent Ji (3x+otdX First set up the appropriate limit u=3x + tramvegge = lim Tazax t> du = 30 x E->00 Juu? 3 du Substitute New limits for new variables X==) u=3.170 =4 \xt=u=3t+| p3t+1 -2 Ju u du = lim t-> t->00 (-1) u 1 - lim t->30 - u Ott es to zero

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