University:
University of Central Florida
Course:
MAC 2312 | Calculus with Analytic Geometry II
Academic year:

2023
Views:

658

Pages:

5
Author:

Maria

3.7. Improper Integrals f(x)= 2 X Sdx A=1 t Instead dx X 1 lim 118 Type I Improper Integral - Domain (x) is infinite § f(x) dx = lim & f(x) dx 6 b b or & f(x)dx = lim f(x)dx ナリー + or 5 f(x)dx = 5 f(x)dx + 5 f(x)dx 100 If limit exists integral converges. If limit ONE integral diverges Ex: Sdx = lim dx = lim In/xl/t 1 = lim Int-In 1 = lim Int = 0∞-) diverges 001- Ex: S f xe dx = lim u= x 0 xe dx dif dv = exdx int du= 1 dx v=e = lim xeli - Sedx = lim Q.e-tet- (e-et) 0 1 0 lim et : 0 08-5-7 lim tet ~-0.0 t-1-00 lim + +- et 18 00 =-1 converges LH: Lim 1 0 0 Ex: ST+xz dx = S +x dx + 8 +x dx 1 1+x2 - 1+x² S 0 xdx = lim dx = lim tan x It 1+x² 0 = lim tan' t - tan'o 1+ x² 0 001-7 0 y=tan = lim tax I+ t S 2 2 I d x = + Ⅱ Ⅲ- Converges to 1 + x2 2 2 858 dx diverges S I dx → converges Ex: For which values of p does 1x converge? dx P X t lim XP t dx = lim xdx lim Glx1/ If p=1 then = lim (n/xl t diverges If p≠ 1 then lim -P+1 100 -p+1 Converge if-p+1 < 0=>p>1 =) power t = lim t-P+1 +∞-p+1 1P+1 -p+1 Type II Improper Integral Range (y) goes to infinity. (Domain of integration contains a vertical asymptate) If vertical asymptote at x = c Sdx = lim dx to t 2 e t § f(x) dx = lim }f(x)dx te a b § f(x)dx = lim of f(x)dx OR OR f(x)dx where acccb C § f(x)dx + 6 $f(x)dx If limit exist converges If any limit DNE => diverges Split at O ble it's the vertical asymptote Ex: Idx= 2 X -2 -2x - dx + f dx 2 Consider x2 lim, & dx = lim l dx = lim = lim + x² +1 t X → oliverges = Sdx also diverges 2 -2 x Ex: S S dx = lim 1+2+ 1 √x-2 de 4-x-2 du: Idx √x-2 3 = lim S t-2+ t-2 I du= lim 2u Ju : converges to 253 +2+ 3 1-2 = lim 253-2+-2 +32+ Comparison Test for Improper Integrals Suppose as f(x) (g(x) for x 2 a Then if I g(x) dx converges then I f(x)dx also converges OL fg(x) dx al20 diverges or if f(x)dx diverges then Ex: Show that dx converges -X Notice ose se for x21 then fodxs fedxs je da dx