Observe lhat ( 4) J '- JR [.\'2] H P.Y. .r dx = Jim x dx =

Chapter 0, Problem 7.1

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Observe lhat ( 4) J '- JR [.\'2] H P.Y. .r dx = Jim x dx = Jim - R x R -x ? '- H - R = Jim 0= 0. R " On 1hc 01her hand. (5) J :x; JO f R, .r d.r = Jim .rd.,. + Jim .r d.r '-' R, :'.\. R1 R:'-.11 [ )]II [ )JR' = Jim x- + lim r - R X ? R' >X ? I . - . R1 . . - II R"!. R"!. J . J J' ) 1111 - + I Ill --: R, -x; 2 R: '- 2 and since lhcse Jasl 1wo Jimics do 1101 exisL we find 1ha11hc improper inlegraJ (5) fails lo ex isl. Bui suppose lhal f (.r) (-:x::i < .r < oc) is an ne11 funclion. one whe~ f(-.r) = f(.r) for all.,. and as.rnme t'1at t'1e Ca11c'1y prindpal m/11e (3) exists. The symmclry or 1he graph or y = /(.r) wilh ~spccl lo 1he y axis lells us ll1at ; .o J JR, f(.r)d.r = - f(.r)d.r . ? . R, - R, and 1 R: J JR: .f(.r)d.r = :;- f(.r)dx. o - R: SEC. 85 EVAU: .. \TIO!'\ or IMPROPER l!\TEGR.-\LS 261 Thus / () 1R~ I IR, I JR~ f (.r) d.r + f (.r) dx = ~ .f dx = ~ [r.v.j" f