# Ln 2 0 ln 2 0 ln 2 ln2 0 0 ln1 2 2 2 1 2 t t t t e e

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()()()ln 20ln 20ln 2ln200ln(1 2)2212tttteedteeeeeee=+=++=+=18.()()66232116431143302512xxdxxxdxxx===20.()()()()04304305326126262261253,8885xdxxdxx+=+++==22.22321232312311(1)1(1)(1)31 (1)31754xdxxxd xx+=+++==24.1122001206(2 )3113ln(1)3ln 2tt dtdtttt=++=+=
200Chapter 5. Integration26.261(1)(2)ttdt+Let2ut=so13tu+=+,dudt=.Further when1t=,1u= −and2t=,0u=. Thus20661107610871(1)(2)(3)(3)3871756ttdtuu duuuduuu+=+=+=+=28.21lneedxxxLetln,.dxux dux==If2,1, and if,2.xe uxeu====221211lnlnln 20.693eedudxxxuu====30.43/ 21(1)xdxxLet1,.2dxuxdux==If1,0, and if4,1.xuxu====43/ 213/ 20115/ 20(1)24545xdxuduxu===32.1311331133[4( )3 ( )]4( )3 ( )4( )3( )4(0)3(4)12==== −fxg xdxfx dxg x dxfx dxg x dx34.3223( )( )5fx dxf x dx= −= −36.Since212331( )( )( )g x dxg x dxg x dx=+wehave221133( )( )( )246 .g x dxg x dxg x dx== −= −38.[]1311332( )3 ( )2( )3( )2(0)3(4)12fxg xdxfx dxg x dx+=+=+=40.4043/ 21/ 2045/ 23/ 20Area(1)()225364165327215x xdxxxdxxx=+=+=+=+=42.083Area92=dxxLetu= 92x, thendu=2dx. Ifx=8,u= 25 and ifx= 0,u= 9, so
Chapter 5. Integration201()()08089259253Area9231( 2)29231232239253( 2)6== −= −= −= −= −=dxxdxxduuu44.230Areaxxedx=Let2ux=then2duxdx=. If0,0xu==and if3,9xu==so22303090909Area1(2 )2121211220.499938xxuuxedxex dxeduee==== −= −+46.221123Area3ln3ln3ln16eedxxxe====48.At 11:00A.M.,2t=, and at 1:00P.M.,4t=. The number of people who enterthe fair between 11:00A.M. and 1:00P.M.is42'( )Nt dt.50.Let( )P tdenote the revenue generated bythe oil aftertmonths. The price per barrelaftertmonths is $95 plus$0.30 for eachmonth or 95 + 0.30t. At 400 barrels permonth, the revenue is changing at the rateof( )400(950.30 )=+P ttfor024t(months). The total revenuewill be2402420( )400(950.30 )0.30400 952\$946,560.=+=+=P tt dttt52.LetA(t) denote the amount of pollution inthe lake aftertyears. LetP(t) denote thepopulation in thousands of people. Thenwe are given20.60.20.5dPttdt=++thousandpeople/yr and5dAdP=units/thousandpeople.By the chain rule225(0.60.20.5)32.5 units/year.

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