÷ƒ’À;è TeX output 1997.05.08:2007‹ÿÿÿÿ y ý£ ? ýä’§*¡óKñ`y cmr10²18.01–UUSolutions,“Problem“Set“8ŽŸ‘Mó !",š cmsy10¸ŽŽŽ‘WxŽ‘[q˲9.5‘UU#14.ŽŸ‘WIf–UUó  b> cmmi10µa“²is“a“pGositivš¸ãe“constan˜t,“sho˜w“thatŽŸ§j’º– Ÿòc‡óú±u cmex10«ZŽŸù<$’ÖAÓµduŽ’ÇsæŸw‰fe(‰ÉŸ UNŸ÷ª«¸pŽ‘UWŸ÷ª«‰fe 4rŸUUµaŸýróÙ“ Rcmr7±2Ž‘µS¸‘8àµuŸýr±2ŽŽŽŽŽŽŽ’ó÷ú²=‘ÇsinŽ‘6ŸûË*ó O!â…cmsy7·±1ŽŸù<$‘¨…µuŽ‘¨…Ÿw‰fe¹~Ÿ (Ö8'aŽŽŽŽ‘%β+‘8àµCŽŸ>‘W²andŽŸµ£’·áÏŸòc‡«ZŽŸù<$’ÏbìµduŽ’Ä¿«Ÿw‰fe 4rŸ (ÖaŸýr±2Ž‘µS²+‘8àµuŸýr±2ŽŽŽŽŽ’èîh²=Ÿù<$‘ã1Ž‘úKŸw‰feI0Ÿ (ÖµaŽŽŽŽ‘ !V²tanŽ‘“ ŸûÞÿ·±1ŽŸù<$‘(-oµuŽ‘(-oŸw‰fe¹~Ÿ (Ö8'aŽŽŽŽ‘1S²+‘8àµC(ã:ŽŸ#R‘Wóò"V cmbx10¹Solution:ŽŽ’ŒŽ*óý': cmti10ºSinc‘ÿ}'eŸù<$‘¢¨µdŽ‘ÇŸw‰fe 뎟 (ÖdxŽŽŽŽ‘ƒ²sinŽ‘סŸûË*·±1Ž‘+>½µx–Dz=“1µ=Ÿö-­«pŽ‘ Ÿö-­‰felnŸ ÒS²1–8ณµxŸýr±2ŽŽŽŽ‘%loº,ŽŸ‘¬Ÿù<$’¢ŒµdŽ’Ÿ=ÍŸw‰fe íðŸ (ÖduŽŽŽŽ’­ ˜²sinŽ’¹P¶ŸûË*·±1Ž’Ä *²(µu=a²)ŽŽ’åÖ÷=ŽŽŸù<$’Ø1Ž’øÑHŸw‰fe7 Ÿ ŒÉŸ÷s0«pŽ‘ Ÿ÷s0‰fe-ŸŒÐ²1–8ณ²(µu=a²)Ÿýr±2ŽŽŽŽŽŽŽ’3¼{¸Ÿù<$‘HÒµdŽ‘lŸw‰fe íðŸ (ÖduŽŽŽŽ‘6²(µu=a²)ŽŽŽ¤‚q’åÖ÷=ŽŽŸù<$’1Ž’øÑHŸw‰fe9}¥Ÿ ŒÉµaŸ÷s0«pŽ‘ Ÿ÷s0‰fe*4tŸŒÐ²1–8ณµuŸýr±2Ž‘|sµ=aŸýr±2ŽŽŽŽŽŽŽŽŽŽŽ¡’åÖ÷²=ŽŽŸù<$’1ã1Ž’øÑHŸw‰feEÁ6Ÿ ŒÉŸ÷s0«pŽ‘ Ÿ÷s0‰fe;Á5ŸŒÐµaŸýr±2Ž›|s²(1–8ณµuŸýr±2Ž˜µ=aŸýr±2Ž˜²)ŽŽŽŽŽŽŽŽŽ¡’åÖ÷=ŽŽŸù<$’1ã1Ž’øÑHŸw‰feEÁ6Ÿ ŒÉŸ÷s0«pŽ‘ Ÿ÷s0‰fe;Á5ŸŒÐµaŸýr±2Ž›|s²(1–8ณµuŸýr±2Ž˜µ=aŸýr±2Ž˜²)ŽŽŽŽŽŽŽŽŽ¡’åÖ÷=ŽŽŸù<$’ –,1Ž’øÑHŸw‰fe(‰ÉŸ UNŸ÷ª«¸pŽ‘UWŸ÷ª«‰fe 4rŸUUµaŸýr±2Ž‘µS¸‘8àµuŸýr±2ŽŽŽŽŽŽŽŽŽŽŸi’%gºsoŽ’¥<Ÿòc‡«ZŽŸù<$’ÀçèµduŽ’²ûŸw‰fe(‰ÉŸ UNŸ÷ª«¸pŽ‘UWŸ÷ª«‰fe 4rŸUUµaŸýr±2Ž‘µS¸‘8àµuŸýr±2ŽŽŽŽŽŽŽŽŽ’åÖ÷²=ŽŽ’÷žsinŽ’å3ŸûË*·±1ŽŸù<$’‚µuŽ’‚Ÿw‰fe¹~Ÿ (Ö8'aŽŽŽŽ’¥²+‘8àµC(ã:ŽŽŽŸ&i‘WºSinc‘ÿ}'eŸù<$‘¢¨µdŽ‘ÇŸw‰fe 뎟 (ÖdxŽŽŽŽ‘ƒ²tanŽ‘!MŸûÞÿ·±1Ž‘-iiµx–Dz=“1µ=²(1–8à+“µxŸûÞÿ±2Ž‘|s²)º,ŽŸ \Ÿù<$’Œ€ãµdŽ’‰¤$Ÿw‰fe íðŸ (ÖduŽŽŽŽ’—oïŸñæ^«ŽŸù<$’ $.²1Ž’Ÿÿ–Ÿw‰feI0Ÿ (ÖµaŽŽŽŽ’¨&¡²tanŽ’¶˜kŸûÞÿ·±1Ž’ÁTß²(µu=a²)Ÿñæ^«ŽŽŽŽ’ê{ ²=ŽŽŸù<$’='1Ž’ýuqŸw‰fe:mŸ (Öµa²(1–8à+“(µu=a²)Ÿýr±2Ž‘|s²)ŽŽŽŽ’;pñ¸Ÿù<$‘HÒµdŽ‘lŸw‰fe íðŸ (ÖduŽŽŽŽ‘6²(µu=a²)ŽŽŽŸµ«’ê{ =ŽŽŸù<$’Ö 1Ž’ýuqŸw‰fe;Á5Ÿ (ÖµaŸýr±2Ž›|s²(1–8à+“µuŸýr±2Ž˜µ=aŸýr±2Ž˜²)ŽŽŽŽŽŽŽŸ‘¬’ê{ =ŽŽŸù<$’ ª1Ž’ýuqŸw‰fe 4rŸ (ÖµaŸýr±2Ž‘µS²+‘8àµuŸýr±2ŽŽŽŽŽŽŽŽŸg’çºsoŽ’²5ŸŸòc‡«ZŽŸù<$’ɶ¼µduŽ’¿{Ÿw‰fe 4rŸ (ÖaŸýr±2Ž‘µS²+‘8àµuŸýr±2ŽŽŽŽŽŽŽ’ê{ ²=ŽŽŸù<$’ýš 1Ž’ýuqŸw‰feI0Ÿ (ÖµaŽŽŽŽ’œ|²tanŽ’FŸûÞÿ·±1ŽŸù<$’!¨•µuŽ’!¨•Ÿw‰fe¹~Ÿ (Ö8'aŽŽŽŽ’*Î&²+‘8àµC(ã:ŽŽŽŸŽA‘M¸ŽŽŽ‘WxŽ‘[q˲9.5‘UU#16.ŽŸ˜6‘WEv‘ÿqÇaluate‘UUŸòc‡«ZŽŸôÜi‘ UV±1ŽŸ@‘ã·±1ŽŽŸù<$‘¾NµdxŽ‘}ÞŸw‰felnŸ (Ö²1–8à+“µxŸýr±2ŽŽŽŽŽ‘3².ŽŽŸ’è1ŽŽŒ‹* y ý£ ? ýä‘W¹Solution:ŽŽŽŸ¡Ë’¡CŸòc‡«ZŽŸôÜi’«C±1ŽŸ@’¦Ñ<·±1ŽŽŸù<$’¼«ûµdxŽ’´k‹Ÿw‰felnŸ (Ö²1–8à+“µxŸýr±2ŽŽŽŽŽŽŽ’Û ,²=ŽŽ’ìÒJtanŽ’ûDŸûÞÿ·±1Ž’«0µx²]ŸûÞÿ±1ŽŸ™á·±1ŽŽŽŽŽŸ•¨’Û ,²=ŽŽ’ìÒJtanŽ’ûDŸûÞÿ·±1Ž’ˆ²(1)–8ณ²tanŽ‘ªªŸûÞÿ·±1Ž‘g²(¸²1)ŽŽŽ¤’Û ,=ŽŽ’ìÒJµš[Ù=²4–8ณ²(¸µ˜=²4)ŽŽŽ¡’Û ,=ŽŽ’ìÒJµ[Ù=²2µ:ŽŽŽŸüÞ‘M¸ŽŽŽ‘WxŽ‘[q˲9.5–UU#17.‘qÇEv‘ÿqÇaluate“Ÿòc‡«ZŽŸù<$‘MµdxŽ‘31Ÿw‰fe(ÁÆŸ UNŸ÷ª«¸pŽ‘UWŸ÷ª«‰fe loŸUU²1–8ณ²4µxŸýr±2ŽŽŽŽŽŽŽ‘:(*².ŽŸ9é‘W¹Solution:ŽŽ’ŒŽ*ºThe–“çsubstitution“µu–Dz=“2µxº,›“çµdu“²=“2‘ª¨µdx˜ºgivesŽŸ¢Æ’©BSŸòc‡«ZŽŸù<$’Å KµdxŽ’¶ /Ÿw‰fe(ÁÆŸ UNŸ÷ª«¸pŽ‘UWŸ÷ª«‰fe loŸUU²1–8ณ²4µxŸýr±2ŽŽŽŽŽŽŽŽŽ’ê(²=ŽŽ’ûÜFŸòc‡«ZŽŸù<$’ˆÎ²2‘ª¨µdxŽ’º"Ÿw‰fe73Ÿ ŒÉ²2Ÿ÷s0«pŽ‘ Ÿ÷s0‰fe(3ŸŒÐ²1–8ณ²(2µx²)Ÿýr±2ŽŽŽŽŽŽŽŽŽŽŽŸq’ê(²=ŽŽ’ûÜFŸòc‡«ZŽŸù<$’¥>µduŽ’º"Ÿw‰fe(Ä(Ÿ UN²2Ÿ÷ª«¸pŽ‘UWŸ÷ª«‰fenПUU²1–8ณµuŸýr±2ŽŽŽŽŽŽŽŽŽŽŽŸ‚i’ê(²=ŽŽŸù<$’ýy1Ž’ýyŸw‰feŸ (Ö2ŽŽŽŽ’íUsinŽ’4sŸûË*·±1Ž’›µu–8à²+“µCŽŽŽŸ¬’ê(²=ŽŽŸù<$’ýy1Ž’ýyŸw‰feŸ (Ö2ŽŽŽŽ’íUsinŽ’4sŸûË*·±1Ž’ðç²(2µx²)–8à+“µC(ã:ŽŽŽ¤• ‘M¸ŽŽŽ‘WxŽ‘[q˲9.7‘UU#13.Ž©sI‘WFindŸù<$‘šµdyŽ‘ˆˆŸw‰fe 뎟 (ÖdxŽŽŽŽ‘üž²if‘UUµy‘"ñ²=‘ÇsinhŽ‘œp(µxŸü^ÿ±3Ž‘|s²).ŽŸ ‘W¹Solution:ŽŽ’ŒŽ*ºSinc‘ÿ}'eŸù<$‘£ÙµdŽ‘ÇŸw‰fe íðŸ (ÖduŽŽŽŽ‘’å²sinhŽ‘&åµu–Dz=“coshŽ‘c‰µuº,–“çthe“chain“rule“givesŽŸŸù<$’•í³µdŽ’“%Ÿw‰fe 뎟 (ÖdxŽŽŽŽ’ Û޲sinhŽ’²°æ(µxŸûÞÿ±3Ž›|s²)–Ç=“coshŽ‘¸á(µxŸûÞÿ±3Ž˜²)Ÿù<$‘ÁµdŽ‘33Ÿw‰fe 뎟 (ÖdxŽŽŽŽ‘ Qô²(µxŸûÞÿ±3Ž˜²)“=“3µxŸûÞÿ±2Ž‘'²coshŽ‘ä(µxŸûÞÿ±3Ž˜²)µ:Ž¡‘M¸ŽŽŽ‘WxŽ‘[q˲9.7‘UU#15.ަ‘WFindŸù<$‘šµdyŽ‘ˆˆŸw‰fe 뎟 (ÖdxŽŽŽŽ‘üž²if‘UUµy‘"ñ²=‘ÇlnŽ‘ o(tanhŽ‘(3µx²)).ŽŸ ‘W¹Solution:ŽŽ’ŒŽ*ºSinc‘ÿ}'eŸù<$‘£ÙµdŽ‘ÇŸw‰fe íðŸ (ÖduŽŽŽŽ‘’å²tanhŽ‘(=‘µu–Dz=“sec¸ãhŽŽ‘㊟û‡±2Ž‘_ýµuº,–“çthe“chain“rule“givesŽŸŸù<$’ž>µdŽ’›bxŸw‰fe 뎟 (ÖdxŽŽŽŽ’©+á²lnŽ’³+àtanhŽ’Ç+ä(3µx²)ŽŽ’ãª=ŽŽŸù<$’c1Ž’ö¤pŸw‰fe&~?Ÿ (ÖtanhŽ‘(3µx²)ŽŽŽŽŸù<$’"d£µdŽ’‰Ÿw‰fe 뎟 (ÖdxŽŽŽŽ’-R~²tanhŽ’AR‚(3µx²)ŽŽŽŸ‘¬’ãª=ŽŽŸù<$’c1Ž’ö¤pŸw‰fe&~?Ÿ (ÖtanhŽ‘(3µx²)ŽŽŽŽ’ ŽÂ¸‘8à²3sec¸ãhŽŽ‘rŸû‡±2Ž‘˜å²(3µx²)ŽŽŽŸŸå’ãª=ŽŽŸù<$’Œs3‘ª¨coshŽ‘œq(3µx²)Ž’ö¤pŸw‰feO겟 .ìsinhŽ‘ÕX(3µx²)‘ª¨coshŽ‘œqŸû‡±2Ž‘ä²(3µx²)ŽŽŽŽŽŽŽŸâ’ãª=ŽŽ’õq=3csc¸ãhŽŽ–r(3µx²)sec¸ãhŽŽ“(3µx²)ŽŽŽŽŸ’è2ŽŽŒ‹ { y ý£ ? ýä‘WºA¾“lso,‘™ínote–e†that“²sinhŽ‘:Þ(2µu²)–BÕ=“2‘ª¨sinhŽ‘€(µu²)‘ª¨coshŽ‘œq(µu²)–e†ºc‘ÿ}'arries“over“without“a“signŽŸ ‘Wchange–“çfr›ÿ}'om“the“r˜e˜gular“trigonometric.‘™–Thusޤ’´mV²3csc¸ãhŽŽ–r(3µx²)sec¸ãhŽŽ“(3µx²)–Ç=“6csc¸ãhŽŽ‘r(6µx²)Ž¡‘Wºas–“çliste›ÿ}'d“in“the“solutions“in“the“b˜ack“of“the“textb˜o˜ok.ŽŸ‘M¸ŽŽŽ‘WxŽ‘[q˲9.7‘UU#20.ŽŸ5£‘WFind‘UUŸòc‡«ZŽŸù<$‘]Ù²sinhŽ‘+31(3µx²)Ž‘31Ÿw‰fe6¨ãŸ (Ö1–8à+“coshŽ‘*©(3µx²)ŽŽŽŽ‘I¹ïµdx².ŽŸÜ ‘W¹Solution:ŽŽ’ŒŽ*ºSubstitute›“çµu–Dz=“1–8à+“coshŽ‘*©(3µx²)º,˜µdu–Dz=“3‘ª¨sinhŽ‘€(3µx²)º,˜soŽŸÃÜŸù<$’Š ²1Ž’Š Ÿw‰feŸ (Ö3ŽŽŽŽ’‘êkŸòc‡«ZŽŸù<$’¤›²3‘ª¨sinhŽ‘€(3µx²)Ž’žÈGŸw‰fe6¨ãŸ (Ö1–8à+“coshŽ‘*©(3µx²)ŽŽŽŽ’ØOµdxŽŽ’í:“²=ŽŽŸù<$’4ä1Ž’4äŸw‰feŸ (Ö3ŽŽŽŽ’ÀŸòc‡«ZŽŸù<$’ðœµduŽ’ðœŸw‰fe íðŸ (Ö‘š9uŽŽŽŽŽŽŽŸ‘¬’í:“²=ŽŽŸù<$’4ä1Ž’4äŸw‰feŸ (Ö3ŽŽŽŽ’ÀlnŽ’¿¸jµu¸j–8à²+“µCŽŽŽŸ¬’í:“²=ŽŽŸù<$’4ä1Ž’4äŸw‰feŸ (Ö3ŽŽŽŽ’ÀlnŽ’h(1–8à+“coshŽ‘*©(3µx²))“+“µC(ã:ŽŽŽŸÍБM¸ŽŽŽ‘WxŽ‘[q˲10.2–UUFind“the“folloš¸ãwing“in˜tegrals:ŽŸ°‰‘RãŒ#19.ŽŽŽ‘mŸòc‡«ZŽŸù<$‘yÝÞµeŸü^ÿ±tanŽ‘ ‡ ŸùZqó °Ü0ncmsy5¶ó†›Zcmr5°1Ž‘6Ÿü^ÿó 0e—rcmmi7´xŽŽ‘yÝÞŸw‰feçuŸ (Ö‘=„²1–8à+“µxŸýr±2ŽŽŽŽŽ’œ£.µdx².ŽŸR‘m¹Solution:ŽŽ’¢Ž*ºSubstitute›“çµu–Dz=“tanŽ‘8âŸüRH·±1Ž‘Ÿþµxº,˜µdu“²=Ÿù<$‘ :»µdxŽ‘úKŸw‰felnŸ (Ö²1–8à+“µxŸýr±2ŽŽŽŽŽ‘ ™ìº:ŽŸ#p ’•ŽÕŸòc‡«ZŽŸù<$’¢l±µeŸü^ÿ±tanŽ‘ ‡ ŸùZq¶°1Ž‘6Ÿü^ÿ´xŽŽ’¢l±Ÿw‰feçuŸ (Ö‘=„²1–8à+“µxŸýr±2ŽŽŽŽŽ’Å2µdx–Dz=“Ÿòc‡«ZŽ‘qÁµeŸûÞÿ´uŽ‘å µdu“²=“µeŸûÞÿ´uŽ‘sB²+›8àµC‘~4²=“µeŸûÞÿ±tanŽ‘ ‡ ŸøÚq¶°1Ž‘6ŸûÞÿ´xŽ‘xT²+˜µC(ã:ŽŸ#8ó‘R㌲#27.ŽŽŽ‘mŸòc‡«ZŽŸù<$’ˆ£3µeŸü^ÿ´xŽŽ‘yÝÞŸw‰fe';ŽŸ UNŸ÷ª«¸pŽ‘UWŸ÷ª«‰feæ7ŸUU²1–8ณµeŸýr±2´xŽŽŽŽŽŽŽ’£÷Gµdx².ŽŸ>‘m¹Solution:ŽŽ’¢Ž*ºSubstitute›“çµu–Dz=“µeŸü^ÿ´xŽ‘äº,˜µdu“²=“µeŸü^ÿ´xŽ‘³Œµdxº:ŽŸŽ@‘p;Ÿòc‡«ZŽŸù<$’×*µeŸü^ÿ´xŽŽ‘}ñŸw‰fe1-VŸ ŒÉŸ÷s0«pŽ‘ Ÿ÷s0‰fe'-UŸŒÐ²1–8ณ²(µeŸýr´xŽ‘ä²)Ÿýr±2ŽŽŽŽŽŽŽ’±$"µdx–Dz=“Ÿòc‡«ZŽŸù<$‘²1Ž‘¤ôŸw‰fe#Ä'Ÿ UNŸ÷ª«¸pŽ‘UWŸ÷ª«‰fenПUU²1–8ณµuŸýr±2ŽŽŽŽŽŽŽ‘6Föµdu“²=“sinŽ‘6ŸûË*·±1Ž‘uRµu–8à²+“µC‘~4²=‘ÇsinŽ‘6ŸûË*·±1Ž‘ʪ²(µeŸûÞÿ´xŽ‘ä²)“+“µC(ã:ŽŸ'ÈX‘R㌲#38.ŽŽŽ‘mŸòc‡«ZŽŸù<$‘|Œ²sinŽ’ˆÓ4ŸûË*·±1Ž’•:PµxŽ‘yÝÞŸw‰fe#ÁÅŸ UNŸ÷ª«¸pŽ‘UWŸ÷ª«‰felnŸUU²1–8ณµxŸýr±2ŽŽŽŽŽŽŽ’ }~µdx².ŽŸ‚i‘m¹Solution:ŽŽ’¢Ž*ºSubstitute›“çµu–Dz=“sinŽ‘6ŸûË*·±1Ž‘uRµxº,˜µdu“²=Ÿù<$‘[-1Ž‘úKŸw‰fe#ÁÅŸ UNŸ÷ª«¸pŽ‘UWŸ÷ª«‰felnŸUU²1–8ณµxŸýr±2ŽŽŽŽŽŽŽ‘*™ëµdxº:ŽŸ#ÈP’ˆÅ9Ÿòc‡«ZŽŸù<$’˜QM²sinŽ’¤˜kŸûË*·±1Ž’°ÿ‡µxŽ’•£Ÿw‰fe#ÁÅŸ UNŸ÷ª«¸pŽ‘UWŸ÷ª«‰felnŸUU²1–8ณµxŸýr±2ŽŽŽŽŽŽŽ’¼Bµµdx–Dz=“Ÿòc‡«ZŽ‘qÁµu‘ª¨du“²=Ÿù<$‘úK1Ž‘úKŸw‰feŸ (Ö2ŽŽŽŽ– -µuŸûÞÿ±2Ž›µS²+‘8àµC‘~4²=Ÿù<$‘úK1Ž‘úKŸw‰feŸ (Ö2ŽŽŽŽ“(sinŽ‘ GŸûË*·±1Ž‘®:µx²)ŸûÞÿ±2Ž˜²+‘8àµC(ã:ŽŽŸ’è²3ŽŽŒ‹ y ý£ ? ýé¦o‘R㌲#45.ŽŽŽ‘mŸòc‡«ZŽŸôÜi‘w±2ŽŸ@‘rŽ<1ŽŽŸù<$’ ‰²(2µx–8à²+“1)‘ª¨µdxŽ‘~ZQŸw‰fe5±¿Ÿ UNŸ÷ª«¸pŽ‘UWŸ÷ª«‰fe-\hŸUUµxŸýr±2Ž‘µS²+–8àµx“²+“2ŽŽŽŽŽŽ’µ?C.ŽŸ‘m¹Solution:ŽŽ’¢Ž*ºSubstitute›lµu–Dz=“µxŸü^ÿ±2Ž‘¢_²+–%ìµx“²+“2º,‘.¸µdu–Dz=“(2µx–%ì²+“1)‘ª¨µdx˜º(note˜the˜newޤ ‘minte–ÿ}'gr“al–“çis“fr‘ÿ}'om“µu–Dz=“1Ÿü^ÿ±2Ž›µS²+–8à1“+“2–Ç=“4–“çºto“µu–Dz=“2Ÿü^ÿ±2Ž˜²+–8à2“+“2–Ç=“8º):ŽŸX’‡RÊŸòc‡«ZŽŸôÜi’‘R˱2ŽŸ@’Œá1ŽŽŸù<$’›_Q²(2µx–8à²+“1)‘ª¨µdxŽ’˜­Ÿw‰fe5±¿Ÿ UNŸ÷ª«¸pŽ‘UWŸ÷ª«‰fe-\hŸUUµxŸýr±2Ž‘µS²+–8àµx“²+“2ŽŽŽŽŽŽ’ÒY#=‘ÇŸòc‡«ZŽŸôÜi‘ DZ8ŽŸ@‘UR4ŽŽ‘î4µuŸûÞÿ·±1´=±2Ž‘|sµdu–Dz=‘úK2µuŸûÞÿ±1´=±2Ž‘ ‘ÊŸôæ`«iŽŽŸöøÝ‘þx±8ŽŸ(‘þx4ŽŽ‘'B²=“4(Ÿ÷=V¸pŽ‘UWŸ÷=V‰feŸª²2ŽŽŽ‘Ž8¸‘8à²1)µ:ŽŸ%¸Á‘R㌲#48.ŽŽŽ‘mŸòc‡«ZŽŸôÜi‘w´@L=±3ŽŸ@‘rŽ<0ŽŽ’†-q²secŽ’“ÆŸûÞÿ±3Ž’™)áµx‘ª¨²tanŽ‘Çµx‘ª¨dx².ŽŸIï‘m¹Solution:ŽŽ’¢Ž*ºSubstitute‘Iµu–ز=“secŽ›C•µxº,‘B!µdu“²=“secŽ˜µx‘ª¨²tanŽ‘Çµx‘ª¨dx–Iº(the“new“inte-Ž¡‘mgr›ÿ}'al–“çis“fr˜om“µu–Dz=“secŽ‘G0“=“1–“çºto“µu–Dz=“secŽ‘œm(µ[Ù=²3)“=“2º)ŽŸ;¦’‘9VŸòc‡«ZŽŸôÜi’›9W´@L=±3ŽŸ@’–Ç0ŽŽ’ªfŲsecŽ’·<ŸûÞÿ±2Ž’½c5µx‘ª¨²secŽ‘*¥µx‘ª¨²tanŽ‘Çµx‘ª¨dx–Dz=“Ÿòc‡«ZŽŸôÜi‘ DZ2ŽŸ@‘UR1ŽŽ‘î4µuŸûÞÿ±2Ž‘'µdu“²=Ÿù<$‘-~1Ž‘-~Ÿw‰feŸ (Ö3ŽŽŽŽ‘ `²µuŸûÞÿ±3Ž‘|sŸñæ^«ŽŽŸóøÛ‘ÝÁ±2ŽŸ,‘ÝÁ1ŽŽ‘"!L²=Ÿù<$‘úK7Ž‘úKŸw‰feŸ (Ö3ŽŽŽŽ‘ -µ:ŽŸú2‘M¸ŽŽŽ‘WxŽ‘[q˲10.3–UUFind“the“folloš¸ãwing“in˜tegrals:ŽŸ;¶‘Wã#2.ŽŽŽ‘mŸòc‡«ZŽ‘xª«²sinŽ’„ñÉŸûË*±4Ž’‹äµx‘ª¨dx².ŽŸž‘m¹Solution:ŽŽ’¢Ž*ºThe–“çformula“²sinŽ‘ÛŸûË*±2Ž‘ µx‘Dz=Ÿü ‘úK±1Ž‘úKŸ£&‰feüsŸ¿˜2ŽŽŽŽ‘ )ñ²(1–8ณ²cosŽ‘œo(2µx²))‘“çºgivesޤÚ’‰ 1²sinŽ’•TOŸûË*±4Ž’›{jµx‘Dz=Ÿù<$‘úK1Ž‘úKŸw‰feŸ (Ö4ŽŽŽŽ› -(1–8ณ²cosŽ‘œo(2µx²))ŸûÞÿ±2Ž‘C‹²=Ÿù<$‘úK1Ž‘úKŸw‰feŸ (Ö4ŽŽŽŽ˜(1“¸“²2‘ª¨cosŽ‘7(2µx²)“+“cosŽ‘œoŸûÞÿ±2Ž‘â²(2µx²))µ:ŽŸ€~‘mºSinc›ÿ}'e‘“ç²cosŽ‘÷vŸü^ÿ±2Ž‘sé²(2µx²)‘Ç=Ÿü ‘úK±1Ž‘úKŸ£&‰feüsŸ¿˜2ŽŽŽŽ‘ )ñ²(1–8à+“cosŽ‘œo(4µx²))º,–“çthis“b˜e˜c˜omesŽ¡Ÿù<$‘wäN²1Ž‘wäNŸw‰feŸ (Ö4ŽŽŽŽ‘~‚(1–8ณ²2‘ª¨cosŽ›7(2µx²)“+Ÿù<$‘l1Ž‘lŸw‰feŸ (Ö2ŽŽŽŽ‘ ŸG(1“+“cosŽ‘œo(4µx²)))‘Ç=Ÿù<$‘úK1Ž‘úKŸw‰feŸ (Ö8ŽŽŽŽ‘ -(3“¸“²4‘ª¨cosŽ˜(2µx²)“+“cosŽ‘œo(4µx²))µ:ŽŸª+‘mºSinc‘ÿ}'e‘“çŸòc‡«ZŽ‘>²cosŽ‘¢(µax²)‘ª¨µdx‘Dz=Ÿù<$‘ã1Ž‘úKŸw‰feI0Ÿ (ÖµaŽŽŽŽ‘ !V²sinŽ‘ht(µax²)–8à+“µC‘·º,–“çwe“getŽŸ jœ’“¤ Ÿòc‡«ZŽ’ŸN²²sinŽ’«•ПûË*±4Ž’±¼ëµx‘ª¨dxŽŽ’Î =²=ŽŽŸù<$’áŽ1Ž’áŽŸw‰feŸ (Ö8ŽŽŽŽ’èâjŸòc‡«ZŽ’òâk²(3–8ณ²4‘ª¨cosŽ‘7(2µx²)“+“cosŽ‘œo(4µx²))‘ª¨µdxŽŽŽŸÒ’Î =²=ŽŽŸù<$’áŽ3Ž’áŽŸw‰feŸ (Ö8ŽŽŽŽ’ç7µx–8ฟù<$‘l²1Ž‘lŸw‰feŸ (Ö4ŽŽŽŽ‘ IïsinŽ‘‘ (2µx²)“+Ÿù<$‘ì1Ž‘lŸw‰fe Ÿ (Ö32ŽŽŽŽ‘IðsinŽ‘‘(4µx²)“+“µC(ã:ŽŽŽŸ €æ‘Wã²#6.ŽŽŽ‘mŸòc‡«ZŽ‘xª«²sinŽ’„ñÉŸûË*±2Ž’‹äµx‘ª¨²cosŽ‘7ŸûÞÿ±5Ž‘5Rµx‘ª¨dx².ŽŸ;·‘m¹Solution:ŽŽ’¢Ž*ºThe–ënsubstitution“µu–e¨²=“sinŽ‘Wnµxº,›Pµdu“²=“cosŽ‘sßµx‘ª¨dxº,˜to‘ÿ}'gether‘ënwithŽŸ ‘mthe–“çidentity“²cosŽ‘÷vŸü^ÿ±2Ž‘‘µx–Dz=“1–8ณ²sinŽ‘þŸûË*±2Ž‘§µx‘“çºgivesŽŸ¿é’…AfŸòc‡«ZŽ’ì²sinŽ’3-ŸûË*±2Ž’£ZHµx‘ª¨²cosŽ‘7ŸûÞÿ±5Ž‘5Rµx‘ª¨dxŽŽ’Ú”²=ŽŽ’ì[&Ÿòc‡«ZŽ’øϲsinŽ’LíŸûË*±2Ž’ tµx²(cosŽ‘ cŸûÞÿ±2Ž‘Šªµx²)ŸûÞÿ±2Ž‘'²cosŽ‘5Rµx‘ª¨dxŽŽŽŸ8ó’Ú”²=ŽŽ’ì[&Ÿòc‡«ZŽ’øϵuŸûÞÿ±2Ž›|s²(1–8ณµuŸûÞÿ±2Ž˜²)ŸûÞÿ±2Ž‘'µduŽŽŽŽŸ’è²4ŽŽŒ‹#Y y ý£ ? ýçœy’Ú”²=ŽŽ’ì[&Ÿòc‡«ZŽ’øϵuŸûÞÿ±6Ž–µS¸›8à²2µuŸûÞÿ±4Ž“²+˜µuŸûÞÿ±2Ž‘'µduŽŽŽ©Ò’Ú”²=ŽŽŸù<$’íŽY1Ž’íŽYŸw‰feŸ (Ö7ŽŽŽŽ’óÁµuŸûÞÿ±7Ž–µS¸Ÿù<$‘l²2Ž‘lŸw‰feŸ (Ö5ŽŽŽŽ› ŸGµuŸûÞÿ±5Ž“²+Ÿù<$‘l1Ž‘lŸw‰feŸ (Ö3ŽŽŽŽ˜µuŸûÞÿ±3Ž“²+‘8àµCŽŽŽŸ¬’Ú”²=ŽŽŸù<$’íŽY1Ž’íŽYŸw‰feŸ (Ö7ŽŽŽŽ’õl5sinŽ’³SŸûË*±7Ž’Únµx–8ฟù<$‘l²2Ž‘lŸw‰feŸ (Ö5ŽŽŽŽ‘ IïsinŽ‘‘ ŸûË*±5Ž›¸(µx“²+Ÿù<$‘l1Ž‘lŸw‰feŸ (Ö3ŽŽŽŽ‘ IïsinŽ‘‘ ŸûË*±3Ž˜µx“²+“µC(ã:ŽŽŽŸ!x‚‘Wã²#9.ŽŽŽ‘mŸòc‡«ZŽ‘xª«Ÿ÷]¸pŽ’Ÿ÷]‰fe¨âŸࣲsinŽ‘ ñƵxŽŽŽ’–SŒ²cosŽ’£·ŸûÞÿ±3Ž’©Þ6µx‘ª¨dx².ŽŸŽA‘m¹Solution:ŽŽ’¢Ž*ºSubstituting›“çµu–Dz=“sinŽ‘¸Þµx˜ºand˜µdu“²=“cosŽ‘ÕOµx‘ª¨dx˜ºgivesŽŸŽ@’†êŸòc‡«ZŽ’‘Ä“Ÿ÷]¸pŽ’šêŸ÷]‰fe¨âŸࣲsinŽ‘ ñƵxŽŽŽ’¯mt²cosŽ’¼ÑŸûÞÿ±3Ž’Âøµx‘ª¨dxŽŽ’ßEp²=ŽŽ’ñ ŽŸòc‡«ZŽ’û ²(sinŽ‘ ñƵx²)ŸûÞÿ±1´=±2Ž‘ ‘ʲ(1–8ณµsinŸûÞÿ±2Ž‘|sµx²)‘ª¨cosŽ‘¸ßµx‘ª¨dxŽŽŽ¤8ó’ßEp²=ŽŽ’ñ ŽŸòc‡«ZŽ’ü·7µuŸûÞÿ±1´=±2Ž‘ ‘ʲ(1–8ณµuŸûÞÿ±2Ž‘|s²)‘ª¨µduŽŽŽ¡’ßEp²=ŽŽ’ñ ŽŸòc‡«ZŽ’û ²(µuŸûÞÿ±1´=±2Ž‘ʪ¸‘8àµuŸûÞÿ±5´=±2Ž‘ ‘ʲ)‘ª¨µduŽŽŽ¦’ßEp²=ŽŽŸù<$’ò?Á2Ž’ò?ÁŸw‰feŸ (Ö3ŽŽŽŽ’ørõµuŸûÞÿ±3´=±2Ž–ʪ¸Ÿù<$‘l²2Ž‘lŸw‰feŸ (Ö7ŽŽŽŽ‘ ŸGµuŸûÞÿ±7´=±2Ž“²+‘8àµCŽŽŽŸ\’ßEp²=ŽŽ’ñ ŽŸñæ^«ŽŸù<$’ùœ5²2Ž’ùœ5Ÿw‰feŸ (Ö3ŽŽŽŽ’zsinŽ’k×µx‘8ฟù<$‘l²2Ž‘lŸw‰feŸ (Ö7ŽŽŽŽ‘ IïsinŽ‘‘ ŸûË*±3Ž‘¸(µxŸñæ^«ŽŽ’K™QŸ÷]¸pŽ’S÷]‰fe¨âŸࣲsinŽ‘ ñƵxŽŽŽ’iÐj²+‘8àµC(ã:ŽŽŽŸ&㟑R㌲#13.ŽŽŽ‘mŸòc‡«ZŽŸôÜi‘w´@L=±4ŽŸ@‘rŽ<0ŽŽ’†-q²secŽ’“ÆŸûÞÿ±4Ž’™)áµx‘ª¨dx².ŽŸ@R‘m¹Solution:ŽŽ’¢Ž*ºSubstituting›µu–e²=“tanŽ‘‚µxº,‘b‰µdu“²=“secŽ‘:åŸü^ÿ±2Ž‘bµx‘ª¨dx˜ºand˜using˜theޤ ‘midentity‘,²secŽ‘pŸü^ÿ±2Ž‘(‹µx–Dz=“1–W:+“tanŽ‘ÉŸüRH±2Ž‘ðµx–,º(note“the“new“limits“of“inte–ÿ}'gr“ation)‘,gives:ŽŸã—’„°Ÿòc‡«ZŽŸôÜi’ް´@L=±4ŽŸ@’Š>W0ŽŽ’ÝŒ²secŽ’ª²áŸûÞÿ±4Ž’°Ùüµx‘ª¨dxŽŽ’Í'N²=ŽŽ’ÞîlŸòc‡«ZŽŸôÜi’èîm´@L=±4ŽŸ@’ä|¦0ŽŽ’öq3²(1–8à+“tanŽ‘ªªŸûÞÿ±2Ž‘Ñŵx²)‘ª¨secŽ‘ýŸûÞÿ±2Ž‘§µx‘ª¨dxŽŽŽŸ¦v’Í'N²=ŽŽ’ÞîlŸòc‡«ZŽŸôÜi’èîm±1ŽŸ@’ä|¦0ŽŽ’íjà²(1–8à+“µuŸûÞÿ±2Ž›|s²)‘ª¨µdu‘Dz=‘úKŸñæ^«Ž‘ V¿µu“²+Ÿù<$‘l1Ž‘lŸw‰feŸ (Ö3ŽŽŽŽ‘ ŸGµuŸûÞÿ±3Ž˜Ÿñæ^«ŽŽ‘6AçŸñæ^ŽŽŸóøÛ‘;‰±1ŽŸ,‘;‰0ŽŽ‘B̲=Ÿù<$‘úK4Ž‘úKŸw‰feŸ (Ö3ŽŽŽŽ‘ -µ:ŽŽŽŸñΑM¸ŽŽŽ‘WxŽ‘[q˲10.3‘UU#26.ŽŸ‘WIf–UUµn“²is“anš¸ãy“pGositiv˜e“in˜teger“¸‘Dz2,“sho˜w“thatŽŸæ3’œ=EŸòc‡«ZŽ’§çî²tanŽ’¶Y¸ŸûÞÿ´nŽ’½uÞµx–ª¨dx‘Dz=Ÿù<$‘úKtanŽ‘lŸüRH´n·±1Ž‘#įµxŽ‘úKŸw‰fe%€Ÿ (Ö‘$n–8ณ²1ŽŽŽŽ‘,çÞ¸‘8àŸòc‡«ZŽ‘ ㉲tanŽ‘USŸûÞÿ´n·±2Ž‘-­íµx“dx:ŽŸÍ ‘W¹Solution:ŽŽ’ŒŽ*ºThe–Ž8identity“²tanŽ‘ŸüRH±2Ž‘'µx–Dz=“secŽ‘œmŸü^ÿ±2Ž‘ȵx¸²1–Ž8ºgives“²tanŽ‘ŸüRH´nŽ‘(µx–Dz=“tanŽ‘8âŸüRH´n·±2Ž‘"‘|µx‘ª¨²secŽ‘ýŸü^ÿ±2Ž‘§µx¸Ž¡‘W²tanŽ‘eqÌŸüRH´n·±2Ž‘u¾º,‘“çsoŽŸŽ@’„ îŸòc‡«ZŽ’Ë—²tanŽ’ž=aŸûÞÿ´nŽ’¥Y‡µx›ª¨dx–Dz=“Ÿòc‡«ZŽ‘qÁ²tanŽ‘㋟ûÞÿ´n·±2Ž‘.<%µx˜²secŽ‘ýŸûÞÿ±2Ž‘§µx˜dx–8ณŸòc‡«ZŽ‘ ㉲tanŽ‘USŸûÞÿ´n·±2Ž‘-­íµx˜dx:ŽŽŸ’è²5ŽŽŒ‹1— y ý£ ? ýä‘WºEvaluating– 8the“ rst“inte–ÿ}'gr“al– 8on“the“right-hand-side“using“the“substitutionޤ ‘Wµu–Dz=“tanŽ‘㊵xº,›“çµdu“²=“secŽ‘œmŸü^ÿ±2Ž‘ȵx‘ª¨dxº,˜we˜getŽŸ²Ã‘j$lŸòc‡«ZŽ‘uϲtanŽ’„@ߟûÞÿ´n·±2Ž’•™yµx‘ª¨²secŽ‘ýŸûÞÿ±2Ž‘§µx‘ª¨dx–Dz=“Ÿòc‡«ZŽ‘qÁµuŸûÞÿ´n·±2Ž‘Xšµdu“²=Ÿù<$‘ãPµuŸü^ÿ´n·±1ŽŽ‘úKŸw‰fe9yŸ (Öµn–8ณ²1ŽŽŽŽ‘Ÿ×+–8àµC‘~4²=Ÿù<$‘úKtanŽ‘lŸüRH´n·±1Ž‘#įµxŽ‘úKŸw‰fe%€Ÿ (Ö‘$n“¸“²1ŽŽŽŽ‘,çÞ+“µC(ã:ŽŸé ‘Wº(We–“çc›ÿ}'an“\absorb"“the“c˜onstant“in“the“se˜c˜ond“inde nite“inte˜gr˜al“ab˜ove.)ŽŸÌ‘M¸ŽŽŽ‘WxŽ‘[q˲10.4–UUFind“the“folloš¸ãwing“in˜tegrals:ŽŸ‘¶‘Wã#2.ŽŽŽ‘mŸòc‡«ZŽŸù<$’€YÞµxŸü^ÿ±2Ž‘'µdxŽ‘yÝÞŸw‰fe#ÁÅŸ UNŸ÷ª«¸pŽ‘UWŸ÷ª«‰felnŸUU²4–8ณµxŸýr±2ŽŽŽŽŽŽŽ’žÒÖ².Ž©‘m¹Solution:ŽŽ’¢Ž*ºWe–ësubstitute“µx–Dz=“2‘ª¨sinŽ‘œnµGº,› Êso‘뵑5²=“sinŽ‘6ŸûË*·±1ŽŸù<$‘¨…µxŽ‘¨…Ÿw‰fe·Ÿ (Ö[޲2ŽŽŽŽ‘#’Ôº,˜µdx“²=“2‘ª¨cosŽ‘¸ßµ‘ñÅdŽŸþ(‘mºand‘“çŸ÷ª«¸pŽ‘ é>Ÿ÷ª«‰felnŸUU²4–8ณµxŸýr±2ŽŽŽŽ‘*IJ=‘Ç2‘ª¨cosŽ‘¸ßµGº.‘™–This‘“çgivesŽŸ4ÛrŸä^o’®@NŸòc‡«ZŽŸù<$’Áš*µxŸü^ÿ±2Ž‘'µdxŽ’»*Ÿw‰fe#ÁÅŸ UNŸ÷ª«¸pŽ‘UWŸ÷ª«‰felnŸUU²4–8ณµxŸýr±2ŽŽŽŽŽŽŽŽ’ê"²=Ž’ûÚ@Ÿòc‡«ZŽŸù<$’¸²4‘ª¨sinŽ‘ ñÆŸûË*±2Ž‘áµ‘ý¸‘8à²2‘ª¨cosŽ‘¸ßµ‘ñÅdŽ’¸Ÿw‰feKÔeŸ (Ö‘‘Q²2‘ª¨cosŽ‘¸ßµŽŽŽŽŽŽŸé?’ê"²=Ž’ûÚ@Ÿòc‡«ZŽ’„é²4‘ª¨sinŽ‘ ñÆŸûË*±2Ž‘áµG;‘ª¨dŽŽŸ8ó’ê"²=Ž’ûÚ@Ÿòc‡«ZŽ’ÚA²(2–8ณ²2‘ª¨cosŽ‘7(2µG²))‘ª¨µdŽŽŸÞ’ê"²=Ž’ûÚ@2µ‘ý¸‘8à²sinŽ‘þ(2µG²)–8à+“µC(ã:ŽŽŽŽŸ5i«‘mºT‘ÿ;¼o–t%expr›ÿ}'ess“²sinŽ‘»C(2µG²)“ºin“terms“of“µxº,‘zwe“c˜ould“use“the“identity“²sinŽ‘»C(2µG²)‘Ç=Ž¡‘m2‘ª¨sinŽ‘œnµ‘ñŲcosŽ‘ÿüµGº,–“çwhich“givesŽŸÐk’›j2²2µ‘ý¸‘8à²sinŽ‘þ(2µG²)–8à+“µC‘~4²=‘Ç2‘ª¨sinŽ‘ ñÆŸûË*·±1ŽŸù<$‘ŒµxŽ‘ŒŸw‰fe·Ÿ (Ö[޲2ŽŽŽŽ‘$¯D¸Ÿù<$‘lµxŽ‘lŸw‰fe·Ÿ (Ö[޲2ŽŽŽŽ‘ VbŸö-­«pŽ‘VcŸö-­‰felnŸ ÒS²4“¸“µxŸýr±2ŽŽŽŽ‘1û±²+“µC(ã:ŽŸºY‘Wã²#6.ŽŽŽ‘mŸòc‡«ZŽŸù<$’‹‡YµdxŽ‘yÝÞŸw‰fe.>ƒŸ UNxŸ÷ª«¸pŽ‘UWŸ÷ª«‰fe 2ŸUUµaŸýr±2Ž‘µS¸‘8àµxŸýr±2ŽŽŽŽŽŽŽŽ’©O”µ:ަ‘m¹Solution:ŽŽ’¢Ž*ºWe–Ë1substitute“µx–Dz=“µa‘ª¨²sinŽ‘œnµGº,›óVso‘Ë1µ‘5²=“sinŽ‘6ŸûË*·±1ŽŸù<$‘¨…µxŽ‘¨…Ÿw‰fe·Ÿ (Ö6öaŽŽŽŽ‘#’Ôº,˜µdx“²=“µa‘ª¨²cosŽ‘¸ßµ‘ñÅdŽŸþ(‘mºand‘“çŸ÷ª«¸pŽ‘ é>Ÿ÷ª«‰fe 2ŸUUµaŸýr±2Ž‘µS¸‘8àµxŸýr±2ŽŽŽŽ‘.âf²=‘ǵa‘ª¨²cosŽ‘¸ßµGº.‘™–This‘“çgivesŽŸsú’™’Ÿòc‡«ZŽŸù<$’¸hµdxŽ’¦oíŸw‰fe.>ƒŸ UNxŸ÷ª«¸pŽ‘UWŸ÷ª«‰fe 2ŸUUµaŸýr±2Ž‘µS¸‘8àµxŸýr±2ŽŽŽŽŽŽŽŽŽŽ’ßᣲ=ŽŽ’ñ¨ÁŸòc‡«ZŽŸù<$’#µa‘ª¨²cosŽ‘¸ßµ‘ñÅdŽ’þ†Ÿw‰fe4 ¸Ÿ (ÖaŸýr±2Ž‘'²sinŽ‘áµ‘ñŲcosŽ‘ÿüµŽŽŽŽŽŽŽŸé?’ßᣲ=ŽŽ’ñ¨ÁŸòc‡«ZŽŸù<$’þ«5²1Ž’þ†Ÿw‰feI0Ÿ (ÖµaŽŽŽŽ’­¨²cscŽ’-¥µ‘ñÅdŽŽŽŸÒ’ßᣲ=ŽŽ’ñ¨Á¸Ÿù<$‘W˲1Ž‘33Ÿw‰feI0Ÿ (ÖµaŽŽŽŽ‘ Z>²lnŽ‘Z=¸j‘ª¨²cscŽ‘*¥µ‘ý²+‘8àcotŽ‘8ßµG¸j–8à²+“µCŽŽŽŸ\’ßᣲ=ŽŽ’ñ¨Á¸Ÿù<$‘W˲1Ž‘33Ÿw‰feI0Ÿ (ÖµaŽŽŽŽ‘ Z>²lnŽŸñø‘Z=« ޤ‘Z= Ž¡‘Z= Ž¡‘Z= ŽŽŸù<$‘âŲ1–8à+“cosŽ‘GµŽ‘âÅŸw‰fe%?úŸ (Ö‘ *¨²sinŽ‘nµŽŽŽŽŸñø‘>Uò« ޤ‘>Uò Ž¡‘>Uò Ž¡‘>Uò ŽŽŽ‘Cä'²+‘8àµCŽŽŽŸ’ßᣲ=ŽŽ’ñ¨Á¸Ÿù<$‘W˲1Ž‘33Ÿw‰feI0Ÿ (ÖµaŽŽŽŽ‘ Z>²lnŽŸîö‘Z=« ޤ‘Z= Ž¡‘Z= Ž¡‘Z= Ž¡‘Z= ŽŽŸø¦Ó‘âŲ1–8à+Ÿü ‘™±1Ž‘lŸ£&‰feVpŸ¿˜´aŽŽŽŽ‘õ¶Ÿ÷ª«¸pŽ‘K Ÿ÷ª«‰fe 2ŸUUµaŸýr±2Ž‘µS¸“µxŸýr±2ŽŽŽŽŽ‘âÅŸ _‰fe@}Ÿ (ÖŸü ‘ú´xŽ‘úŸ£&‰feˆäŸ¿˜:aŽŽŽŽŽŽŽŸîö‘Y“« ޤ‘Y“ Ž¡‘Y“ Ž¡‘Y“ Ž¡‘Y“ ŽŽŽ‘_!I²+‘8àµCŽŽŽŸ"’ßᣲ=ŽŽ’ñ¨Á¸Ÿù<$‘W˲1Ž‘33Ÿw‰feI0Ÿ (ÖµaŽŽŽŽ‘ Z>²lnŽŸîö‘Z=« ޤ‘Z= Ž¡‘Z= Ž¡‘Z= Ž¡‘Z= ŽŽŸù<$‘âŵa–8à²+“Ÿ÷ª«¸pŽ‘ Ž7Ÿ÷ª«‰fe 2ŸUUµaŸýr±2Ž‘µS¸“µxŸýr±2ŽŽŽŽŽ‘âÅŸw‰fe: uŸ (Ö‘)-µxŽŽŽŽŸîö‘Sm« ޤ‘Sm Ž¡‘Sm Ž¡‘Sm Ž¡‘Sm ŽŽŽ‘X­¢²+‘8àµC(ã:ŽŽŽŽŸ’è²6ŽŽŒ‹>] y ý£ ? ýçµ£‘R㌲#14.ŽŽŽ‘mŸòc‡«ZŽŸù<$’Ž­“µdxŽ‘yÝÞŸw‰fe4ŠøŸ V½²(µaŸýr±2Ž‘µS¸‘8àµxŸýr±2Ž‘|s²)Ÿýr±3´=±2ŽŽŽŽŽ’¯œ µ:ޤ±¼‘m¹Solution:ŽŽ’¢Ž*ºA‘ÿ}'gain–5 we“use“µx–Dz=“µa‘ª¨²sinŽ‘œnµGº,›Hso‘5 µ‘5²=“sinŽ‘6ŸûË*·±1ŽŸù<$‘¨…µxŽ‘¨…Ÿw‰fe·Ÿ (Ö6öaŽŽŽŽ‘#’Ôº,˜µdx“²=“µa‘ª¨²cosŽ‘¸ßµ‘ñÅdŽ©þ(‘mºand‘“çŸ÷ª«¸pŽ‘ é>Ÿ÷ª«‰fe 2ŸUUµaŸýr±2Ž‘µS¸‘8àµxŸýr±2ŽŽŽŽ‘.âf²=‘ǵa‘ª¨²cosŽ‘¸ßµGº.‘™–This‘“çgivesŽŸ]¸’®iâŸòc‡«ZŽŸù<$’ÐsµdxŽ’»G¾Ÿw‰fe4ŠøŸ V½²(µaŸýr±2Ž‘µS¸‘8àµxŸýr±2Ž‘|s²)Ÿýr±3´=±2ŽŽŽŽŽŽŽ’ûé²=ŽŽ’ ÍŸòc‡«ZŽŸù<$’ªãµa‘ª¨²cosŽ‘¸ßµ‘ñÅdŽ’ªãŸw‰fe&ÒñŸ (Ö‘oŒaŸýr±3Ž‘'²cosŽ‘ŠªŸýr±3Ž‘±ÅµŽŽŽŽŽŽŽŸ&i’ûé²=ŽŽ’ ÍŸòc‡«ZŽŸù<$’ ´²1Ž’ªãŸw‰fe Å£Ÿ (ÖµaŸýr±2ŽŽŽŽŽ’&Na²secŽ’3#¶ŸûÞÿ±2Ž’9Jѵ‘ñÅdŽŽŽŸÒ’ûé²=ŽŽŸù<$’c 1Ž’:Ÿw‰fe Å£Ÿ (ÖµaŸýr±2ŽŽŽŽŽ’£¸²tanŽ’*À*µ‘ý²+‘8àµCŽŽŽŸ‘¬’ûé²=ŽŽŸù<$’F˜sinŽ’"8^µŽ’:Ÿw‰fewfŸ (ÖaŸýr±2Ž‘'²cosŽ‘5RµŽŽŽŽ’0ã³²+‘8àµCŽŽŽŸî’ûé²=ŽŽŸù<$’$K1µxŽ’:Ÿw‰fe2M Ÿ UNaŸýr±2Ž‘|sŸ÷ª«¸pŽ‘ ÑÊŸ÷ª«‰fe 2ŸUUµaŸýr±2Ž‘µS¸‘8àµxŸýr±2ŽŽŽŽŽŽŽŽ’C¹W²+‘8àµCŽŽŽŸ!C“‘R㌲#18.ŽŽŽ‘mŸòc‡«ZŽŸù<$’Ž­“µdxŽ‘yÝÞŸw‰fe4ŠøŸ V½²(µxŸýr±2Ž‘µS¸‘8àµaŸýr±2Ž‘|s²)Ÿýr±3´=±2ŽŽŽŽŽ’¯œ µ:Ž¡‘m¹Solution:ŽŽ’¢Ž*ºNow–Ž8we“use“µx–Dz=“µa‘ª¨²secŽ‘*¥µGº,›µ‘5²=“secŽ‘œmŸü^ÿ·±1ŽŸù<$‘6¼µxŽ‘6¼Ÿw‰fe·Ÿ (Ö6öaŽŽŽŽ‘$! º,˜µdx“²=“µa‘ª¨²secŽ‘*¥µ‘ñŲtanŽ‘7µ‘ñÅdަ‘mºand‘“çŸ÷ª«¸pŽ‘ é>Ÿ÷ª«‰fe 2ŸUUµxŸýr±2Ž‘µS¸‘8àµaŸýr±2ŽŽŽŽ‘.âf²=‘ǵa‘ª¨²tanŽ‘ÇµGº.‘™–This‘“çgivesޤ]¸’ŽŠÓŸòc‡«ZŽŸù<$’°8dµdxŽ’›h¯Ÿw‰fe4ŠøŸ V½²(µxŸýr±2Ž‘µS¸‘8àµaŸýr±2Ž‘|s²)Ÿýr±3´=±2ŽŽŽŽŽ’Óíò²=‘ÇŸòc‡«ZŽŸù<$‘¤ôµa‘ª¨²secŽ‘*¥µ‘ñŲtanŽ‘7µ‘ñÅdŽ‘¤ôŸw‰fe=µŸ c¿‘ QaŸýr±3Ž‘'²tanŽ‘˜åŸüRH±3Ž‘ÀµŽŽŽŽ‘P£ô²=Ÿù<$‘]1Ž‘úKŸw‰fe Å£Ÿ (ÖµaŸýr±2ŽŽŽŽŽ‘ÉŸòc‡«ZŽŸù<$‘{¥²cosŽ‘,‰Üµ‘ñÅdŽ‘{¥Ÿw‰feߟ êÝ‘;þ²sinŽ‘ƒŸûË*±2Ž‘ª7µŽŽŽŽ‘>ñ:ŽŸ2‘mºExpr›ÿ}'essing–a9the“inte˜gr˜and“this“way“suggests“the“substitution“µu–Dz=“sinŽ‘¸ÞµGº,Ž© ‘mµdu–Dz=“cosŽ‘ÕOµ‘ñÅdGº,–“çwhich“givesŽ¡Ÿù<$’‰å%²1Ž’‡‚TŸw‰fe Å£Ÿ (ÖµaŸýr±2ŽŽŽŽŽ’”%ÒŸòc‡«ZŽŸù<$’¡®²cosŽ’°åµ‘ñÅdŽ’¡®Ÿw‰feߟ êÝ‘;þ²sinŽ‘ƒŸûË*±2Ž‘ª7µŽŽŽŽ’Äݲ=Ÿù<$‘]1Ž‘úKŸw‰fe Å£Ÿ (ÖµaŸýr±2ŽŽŽŽŽ‘ÉŸòc‡«ZŽ‘HrµuŸûÞÿ·±2Ž‘ gµdu‘Dz=Ÿù<$‘VL¸²1Ž‘úKŸw‰fe!Ÿ (ÖµuaŸýr±2ŽŽŽŽŽ‘å²+–8àµC‘~4²=Ÿù<$‘ Ä6¸²1Ž‘úKŸw‰feZõŸ (ÖµaŸýr±2Ž‘'²sinŽ‘áµŽŽŽŽ‘%ÁS²+“µC(ã:ŽŸD‘mºWe–* just“have“to“expr›ÿ}'ess“²sinŽ‘Òµ‘q)ºin“terms“of“µxº.‘vMR˜e˜ading“this“o “fr˜om“theަ‘mtriangle–h˜with“side-lengths“µaº,‘qBµx–Dz=“µa‘ª¨²secŽ‘*¥µ›¯µºand‘h˜Ÿ÷ª«¸pŽ‘ ½ïŸ÷ª«‰fe 2ŸUUµxŸýr±2Ž‘µS¸‘8àµaŸýr±2ŽŽŽŽ‘.·²=“µa‘ª¨²tanŽ‘Çµ˜ºgivesަ‘m²sinŽ‘zñȵ‘5²=›Ç(µa‘ª¨²tanŽ‘Çµ•G²)µ=²(µa‘ª¨²secŽ‘*¥µ“²)˜=˜Ÿ÷ª«¸pŽ‘ oŸ÷ª«‰fe 2ŸUUµxŸýr±2Ž‘µS¸‘8àµaŸýr±2ŽŽŽŽ‘+Nµ=xº,‘“çsoŽŸëñ’´†XŸòc‡«ZŽŸù<$’Ö3éµdxŽ’Ád4Ÿw‰fe4ŠøŸ V½²(µxŸýr±2Ž‘µS¸‘8àµaŸýr±2Ž‘|s²)Ÿýr±3´=±2ŽŽŽŽŽ’ùéw²=Ÿù<$‘a³¸µxŽ‘úKŸw‰fe2M Ÿ UNaŸýr±2Ž‘|sŸ÷ª«¸pŽ‘ ÑÊŸ÷ª«‰fe 2ŸUUµxŸýr±2Ž‘µS¸‘8àµaŸýr±2ŽŽŽŽŽŽŽŽ‘9³h²+‘8àµC(ã:ŽŸ¼á‘M¸ŽŽŽ‘WxŽ‘[q˲10.4‘UU#31.ŽŸŠÜ‘WIf–¯³the“circle“(µx–u¸“µb²)Ÿü^ÿ±2Ž‘ñ‘²+“µy[ÙŸü^ÿ±2Ž‘6²=–]´µaŸü^ÿ±2Ž‘,&²(0“µ<“a“<“b²)–¯³is“rev•¸ãolv“ed–¯³abGout“the“µy[Ù²-axis,ަ‘Wthe–;`resulting“solid“of“rev¸ãolution“is“called“a“torus.‘i Use“the“shell“methoGd“toަ‘W nd–UUthe“v¸ãolume“of“this“torus.ŽŸ¶N‘W¹Solution:ŽŽ’ŒŽ*ºWe›|·inte–ÿ}'gr“ate˜µdV‘¥¼²=‘lØ2µ[Ùxh‘ª¨dx˜ºfr“om˜µx–lز=“µb–ᓸ“µa˜ºto˜µx–lز=“µb–ᓲ+“µaº,ަ‘Wwher›ÿ}'e–“çµh“ºis“the“height“of“the“cylindric˜al“shel‘‚Øl,“so“µh–Dz=“2Ÿ÷s0«pŽ‘ Ÿ÷s0‰fe8|¶ŸŒÐµaŸýr±2Ž‘µS¸–8à²(µx“¸“µb²)Ÿýr±2ŽŽŽŽ‘B|·º.‘™–SoŽŸÃ(’­QµV‘ÿü²=‘Ç4µ‘Ÿòc‡«ZŽŸôÜi‘ ‚´b±+´aŽŸ@‘”»b·´aŽŽ‘/nµxŸöö5«pŽ‘ Ÿöö5‰fe8|¶Ÿ ˵aŸýr±2Ž‘µS¸–8à²(µx“¸“µb²)Ÿýr±2ŽŽŽŽ‘D'_µdx:ŽŽŸ’è²7ŽŽŒ‹N: y ý£ ? ýä‘WºThis–èmsuggests“substituting“so“that“µx–v¸“µb–`6²=“µa‘ª¨²sinŽ›œnµGº,–ýŽi.e.,“µx–`6²=“µb–v²+“µa‘ª¨²sinŽ˜µGº,‘ýŽsoŽŸ§j‘Wµ‘5²=‘ÇsinŽ‘6ŸûË*·±1ŽŸù<$‘¨…µx–8ณµbŽ‘¨…Ÿw‰fe:¤Ÿ (Ö‘xºaŽŽŽŽ‘4\º,›“çµdx–Dz=“µa‘ª¨²cosŽ‘¸ßµ‘ñÅd‘Ûºand˜Ÿ÷s0«pŽ‘ “èŸ÷s0‰fe8|¶ŸŒÐµaŸýr±2Ž‘µS¸–8à²(µx“¸“µb²)Ÿýr±2ŽŽŽŽ‘H×¶²=“µa‘ª¨²cosŽ‘¸ßµGº.‘™–This˜givesŽŸ:«‘{N»µVŽ’\ô²=ŽŽ’Ÿ$4µ‘Ÿòc‡«ZŽŸôÜi‘ ‚´@L=±2ŽŸ@‘”»·´@L=±2ŽŽ‘W‚²(µb–8à²+“µa‘ª¨²sinŽ‘œnµG²)µaŸûÞÿ±2Ž‘'²cosŽ‘ŠªŸûÞÿ±2Ž‘±Åµ‘ñÅdŽŽŽŸ@’\ô²=ŽŽ’Ÿ$4µ•[ÙaŸûÞÿ±2Ž‘|sµb‘ª¨Ÿòc‡«ZŽŸôÜi‘ ª©´@L=±2ŽŸ@‘8â·´@L=±2ŽŽ‘¦Q²cosŽ‘* àŸûÞÿ±2Ž‘00ûµ›ñÅd‘ý²+‘8à4µ“aŸûÞÿ±3Ž‘'Ÿòc‡«ZŽŸôÜi‘'´@L=±2ŽŸ@‘ µU·´@L=±2ŽŽ‘!"IJsinŽ‘/е˜²cosŽ‘UTŸûÞÿ±2Ž‘|oµ˜dG:ŽŽŽŸJP‘WºWe–±evaluate“the“ rst“inte–ÿ}'gr“al–±using“the“identity“²cosŽ‘€@Ÿü^ÿ±2Ž‘§[µ‘²=Ÿü ‘ò±1Ž‘òŸ£&‰feüsŸ¿˜2ŽŽŽŽ‘ !¾²(1–›ÿ+“cosŽ‘ÿŽ(2µG²))º,Ž© ‘WgivingޤPi‘n8¡µ[ÙaŸûÞÿ±2Ž‘|sµb‘ª¨Ÿòc‡«ZŽŸôÜi‘ ª©´@L=±2ŽŸ@‘8â·´@L=±2ŽŽ‘û©²(1–8à+“cosŽ‘œo(2µG²))‘ª¨2µdŽŽ’ðOC²=ŽŽ’aµ[ÙaŸûÞÿ±2Ž‘|sµb‘ª¨Ÿòc‡«ZŽŸôÜi‘ ª©´ŽŸ@‘8â·´ŽŽ‘æR²(1–8à+“cosŽ‘Gµu²)‘ª¨µduŽŽŽŸ€R’ðOC²=ŽŽ’aµš[ÙaŸûÞÿ±2Ž‘|sµb²(µu–8à²+“sinŽ‘*¦µu²)]ŸûÞÿ´ŽŸ™á·´ŽŽ‘tˆ²=‘Ç2µ˜ŸûÞÿ±2Ž‘ØLµaŸûÞÿ±2Ž‘|sµb:ŽŽŽŸû ‘WºF‘ÿ;¼or–“çthe“se–ÿ}'c“ond›“çinte“gr“al,˜we˜substitute˜µv–"ñ²=‘ÇcosŽ‘ÕOµGº,˜µdv“²=‘Ǹ‘ª¨²sinŽ‘œnµ‘ñÅdGº,˜soŽ¡’ŽwÖ²4µ•[ÙaŸûÞÿ±3Ž‘'Ÿòc‡«ZŽŸôÜi‘'´@L=±2ŽŸ@‘ µU·´@L=±2ŽŽ‘!"IJsinŽ‘/е‘ñŲcosŽ‘UTŸûÞÿ±2Ž‘|oµ‘ñÅd‘5²=›Ç4µ“aŸûÞÿ±3Ž‘'Ÿòc‡«ZŽŸôÜi‘'±0ŽŸ@‘ µU0ŽŽ‘N7¸µv“ŸûÞÿ±2Ž‘‚ôµdv‘"ñ²=˜0µ:ŽŸû]‘WºTher–ÿ}'efor“e–“çthe“volume“µV‘Ì˺is“²2µ[ÙŸü^ÿ±2Ž‘ØLµaŸü^ÿ±2Ž‘|sµbº.ޤû ‘M¸ŽŽŽ‘WxŽ‘[q˲10.4‘UU#33.ŽŸei‘WFind–UUthe“length“of“the“curvš¸ãe“µy‘"ñ²=‘ÇlnŽ‘ ǵx“²bGet˜w˜een“µx–Dz=“1–UUand“µx–Dz=“Ÿ÷ºP¸pŽ‘ oŸ÷ºP‰feŸE°²8ŽŽŽ‘p.Ž¡‘W¹Solution:ŽŽ’ŒŽ*ºWe‘“çinte–ÿ}'gr“ateŽŸꇑb`!µds–Dz=“Ÿöö5«pŽ‘ ÇŸöö5‰fe7ßiŸ ˲1–8à+“(µdy[Ù=dx²)Ÿýr±2ŽŽŽŽ‘FQ*µdx“²=“ŸðId«rŽ‘ ÇŸðId‰feÒÔŸ¶œ²1‘8à+Ÿù<$‘Ú1Ž‘lŸw‰fe 3Ÿ (ÖµxŸýr±2ŽŽŽŽŽŽŽŽ›,D•µdx“²=“ŸïÏ«rŽ‘ ÇŸïωfeÒÔŸ0ëŸù<$‘33µxŸýr±2Ž‘µS²+‘8à1Ž‘33Ÿw‰felnŸ (Ö‘œpµxŸýr±2ŽŽŽŽŽŽ˜µdx“²=Ÿù<$‘úKŸ÷ª«¸pŽ‘ O¢Ÿ÷ª«‰felnŸUUµxŸýr±2Ž‘µS²+‘8à1ŽŽŽ‘úKŸw‰fe#ÁÅŸ (Ö‘UµxŽŽŽŽ‘*™ëdxŽŸð“‘Wºfr‘ÿ}'om›©µx–½9²=“1˜ºto˜µx“²=“Ÿ÷ºP¸pŽ‘ Ÿ÷ºP‰feŸE°²8ŽŽŽ‘‘º.‘Ù!We˜ rst˜ nd˜the˜inde nite˜inte–ÿ}'gr“al˜using˜theަ‘Wsubstitution›˜µx–Dz=“tanŽ‘㊵Gº,‘Äuso˜µ‘5²=“tanŽ‘8âŸüRH·±1Ž‘Ÿþµxº,‘Äuµdx“²=“secŽ‘œmŸü^ÿ±2Ž‘ȵ‘ñÅd‘×µºand˜Ÿ÷ª«¸pŽ‘ åïŸ÷ª«‰felnŸUUµxŸýr±2Ž‘µS²+‘8à1ŽŽŽ‘)u=“secŽ‘GµGº.ަ‘WThis‘“çgivesŽŸÒÍ’ŒvŸòc‡«ZŽŸù<$’™SáŸ÷ª«¸pŽ’¡©8Ÿ÷ª«‰felnŸUUµxŸýr±2Ž‘µS²+‘8à1ŽŽŽ’™SáŸw‰fe#ÁÅŸ (Ö‘UµxŽŽŽŽ’¿ódxŽŽ’Ôß²=ŽŽ’æ¦-Ÿòc‡«ZŽŸù<$’ó„ ²secŽ’µ‘ý¸‘8à²secŽ‘5Ÿü^ÿ±2Ž‘5Pµ‘ñÅdŽ’ó„ Ÿw‰fe>Ÿ (Ö‘´Ý²tanŽ‘$ÑOµŽŽŽŽŽŽŽŸæ’Ôß²=ŽŽ’æ¦-Ÿòc‡«ZŽŸù<$’ó„ ²secŽ’µG²(tanŽ‘qÊŸüRH±2Ž‘˜åµ‘ý²+‘8à1)‘ª¨µdŽ’ó„ Ÿw‰feQ⥟ (Ö‘f¨²tanŽ‘.ƒµŽŽŽŽŽŽŽŸ8ó’Ôß²=ŽŽ’æ¦-Ÿòc‡«ZŽ’ð¦.²(secŽ‘ýµ‘ñŲtanŽ‘7µ‘ý²+‘8àcscŽ‘¸ÝµG²)‘ª¨µdŽŽŽŸz’Ôß²=ŽŽ’æ¦-secŽ’õ&*µ–ý¸‘8à²lnŽ‘ 8߸j‘ª¨²cscŽ‘*¥µ“²+‘8àcotŽ‘8ßµG¸j–8à²+“µC(ã:ŽŽŽŸ0®‘WºT‘ÿ;¼o–áexpr‘ÿ}'ess“this“in“terms“of“µxº,‘ôguse“²cscŽ‘aµ›š²=Ÿù<$‘T`secŽ‘Ô]µŽ‘†%Ÿw‰feVŸ (Ö²tanŽ‘rµŽŽŽŽ‘¯Èºand“²cotŽ‘áµ˜²=Ÿù<$‘ Ð1Ž‘†%Ÿw‰feVŸ (ÖtanŽ‘rµŽŽŽŽ‘ήº.‘/ThisŽŸiÁ‘Wgives:ŽŸqÑ’…=Ÿòc‡«ZŽŸù<$’’[Ÿ÷ª«¸pŽ’šp²Ÿ÷ª«‰felnŸUUµxŸýr±2Ž‘µS²+‘8à1ŽŽŽ’’[Ÿw‰fe#ÁÅŸ (Ö‘UµxŽŽŽŽ’¸ºûdx–Dz==“Ÿö-­«pŽ‘ ÇŸö-­‰felnŸ ÒSµxŸýr±2Ž‘µS²+‘8à1ŽŽŽ‘*lg¸‘8à²lnŽŸîö‘ 8ß« ޤ‘ 8ß Ž¡‘ 8ß Ž¡‘ 8ß Ž¡‘ 8ß ŽŽŸù<$‘ÁgŸ÷ª«¸pŽ‘¾Ÿ÷ª«‰felnŸUUµxŸýr±2Ž‘µS²+‘8à1ŽŽŽ‘6¼ +‘8à1Ž‘ÁgŸw‰fe4ú¤Ÿ (Ö‘¡ÄµxŽŽŽŽŸîö‘Fï>« ޤ‘Fï> Ž¡‘Fï> Ž¡‘Fï> Ž¡‘Fï> ŽŽŽ‘L}s²+‘8àµC(ã:ŽŽŸ’è²8ŽŽŒ‹ ^Ì y ý£ ? ýä‘WºSubtr›ÿ}'acting–“çthe“value“at“µx–Dz=“1–“çºfr˜om“the“value“at“µx–Dz=“Ÿ÷ºP¸pŽ‘ oŸ÷ºP‰feŸE°²8ŽŽŽ‘°Wºgivesޤùí‘u¶Hµs–Dz=“3–8ณ²lnŽ‘ 8ߟ÷=V¸pŽ‘Ž6Ÿ÷=V‰feŸª²2ŽŽŽ‘Ç¸“Ÿ÷=VpŽ‘ Ž7Ÿ÷=V‰feŸª²2ŽŽŽ‘Ç+“lnŽ› Ž7(Ÿ÷=V¸pŽ‘UWŸ÷=V‰feŸª²2ŽŽŽ‘Ž8+“1)–Ç=“3–8ณŸ÷=VpŽ˜Ÿ÷=V‰feŸª²2ŽŽŽ‘Ç+“lnŽ˜(1“+“Ÿ÷=V¸pŽ˜Ÿ÷=V‰feŸª²2ŽŽŽ‘Ž8µ=²2)µ:ŽŸŃ‘M¸ŽŽŽ‘WxŽ‘[q˲10.5–UUCalculate“the“folloš¸ãwing“in˜tegrals:ŽŸ>–‘Wã#2.ŽŽŽ‘mŸòc‡«ZŽŸù<$’‘À÷µdxŽ‘yÝÞŸw‰fe:±ÀŸ UNŸ÷ª«¸pŽ‘UWŸ÷ª«‰fe2\iŸUU²5–8à+“4µx“¸“µxŸýr±2ŽŽŽŽŽŽŽ’µÂѲ.ީչ‘m¹Solution:ŽŽ’¢Ž*ºWe–“çc›ÿ}'omplete“the“squar˜e“by“writingŽ¡‘m²5–Þò+“4µx“¸“µxŸûÞÿ±2Ž›C‹²=‘Ç5“+“4“¸“²4“+“4µx“¸“µxŸûÞÿ±2Ž˜²=›Ç9“¸“²(4“¸“²4µx“²+“µxŸûÞÿ±2Ž‘|s²)˜=˜9“¸“²(µx“¸“²2)ŸûÞÿ±2Ž‘|sµ:Ž¡‘mºThe–“çsubstitution“µu–Dz=“µx–8ณ²2º,›“çµdu–Dz=“µdx˜ºgivesŽŸˆ-‘mŸòc‡«ZŽ‘xª«µdxŽ’ƒ–9Ÿöö5«pŽ’–:Ÿöö5‰fe4lkŸ ˲9–8ณ²(µx“¸“²2)Ÿýr±2ŽŽŽŽŽ’Äɽ²=‘ÇŸòc‡«ZŽ‘qÁµduŽ‘_±Ÿö-­«pŽ‘#_²Ÿö-­‰fenП ÒS²9–8ณµuŸýr±2ŽŽŽŽŽ‘A•š²=‘ÇsinŽ‘6ŸûË*·±1ŽŸù<$‘¨…µuŽ‘¨…Ÿw‰fe¹~Ÿ (Ö\¿²3ŽŽŽŽ‘#•6+µC‘~4²=‘ÇsinŽ‘6ŸûË*·±1ŽŸù<$‘¨…µx–8ณ²2Ž‘¨…Ÿw‰feïûŸ (Ö‘÷ý3ŽŽŽŽ‘4˳+µC(ã:ŽŸ#I²‘Wã²#7.ŽŽŽ‘mŸòc‡«ZŽŸù<$’ƒ5lµxŸü^ÿ±2Ž‘'µdxŽ‘yÝÞŸw‰fe)x០UNŸ÷ª«¸pŽ‘UWŸ÷ª«‰fe!#ŠŸUU²6µx–8ณµxŸýr±2ŽŽŽŽŽŽŽ’¤‰ò².ަ‘m¹Solution:ŽŽ’¢Ž*ºComplete–“çthe“squar‘ÿ}'e“by“writingŽ¡’¤vM²6µx–8ณµxŸûÞÿ±2Ž›C‹²=‘Ç9“¸“²9“+“6µx“¸“µxŸûÞÿ±2Ž˜²=‘Ç3ŸûÞÿ±2Ž‘µS¸“²(µx“¸“²3)ŸûÞÿ±2Ž‘|sµ:Ž¡‘mºNow–òïsubstitute“so“that“µx–7=¸“²3–C=“3‘ª¨sinŽ‘œnµšGº,–J±i.e.,“µx–C²=“3(1–7=+“sinŽ‘)µ˜²)º,‘J±soŽŸµ£‘mµ‘žå²=‘WÈsinŽ‘žæŸûË*·±1ŽŸù<$‘95µx–8ณ²3Ž‘95Ÿw‰feïûŸ (Ö‘÷ý3ŽŽŽŽ‘5\cº,‘÷½µdx–WȲ=“3‘ª¨cosŽ‘¸ßµ‘ñÅd‘*âºand‘ãÅŸ÷ª«¸pŽ‘ 9Ÿ÷ª«‰fe!#ŠŸUU²6µx–8ณµxŸýr±2ŽŽŽŽ‘0´n²=“Ÿ÷s0«pŽ‘ WÉŸ÷s0‰fe8èÞŸŒÐ²3Ÿýr±2Ž‘µS¸–8à²(µx“¸“²3)Ÿýr±2ŽŽŽŽ‘I˜o²=ŽŸÍБm3‘ª¨cosŽ‘¸ßµGº.‘™–This‘“çgivesŽŸÓi’†:»Ÿòc‡«ZŽŸù<$’¦éëµxŸü^ÿ±2Ž‘'µdxŽ’“—Ÿw‰fe>llŸ ŒÉŸ÷s0«pŽ‘ Ÿ÷s0‰fe4lkŸŒÐ²9–8ณ²(µx“¸“²3)Ÿýr±2ŽŽŽŽŽŽŽŽŽ’Ü¸6²=ŽŽ’îTŸòc‡«ZŽŸù<$’û]0²9(1–8à+“sinŽ‘*¦µG²)Ÿü^ÿ±2Ž‘µS¸“²3‘ª¨cosŽ‘¸ßµ‘ñÅdŽ’û]0Ÿw‰fec)ºŸ (Ö‘$;û²3‘ª¨cosŽ‘¸ßµŽŽŽŽŽŽŽŸéG’ܸ6²=ŽŽ’îT9‘ª¨Ÿòc‡«ZŽ‘ ª©²(1–8à+“sinŽ‘*¦µG²)ŸûÞÿ±2Ž‘'µdŽŽŽ¤8ó’ܸ6²=ŽŽ’îT9‘ª¨Ÿòc‡«ZŽ‘ ª©²(1–8à+“2‘ª¨sinŽ‘œnµ‘ý²+“sinŽ‘þŸûË*±2Ž‘§µG²)‘ª¨µdŽŽŽ¡’ܸ6²=ŽŽ’îT9‘ª¨Ÿòc‡«ZŽ‘ ª©²(Ÿù<$‘333Ž‘33Ÿw‰feŸ (Ö2ŽŽŽŽ‘ ŸG+‘8à2‘ª¨sinŽ‘œnµ‘ý¸Ÿù<$‘l²1Ž‘lŸw‰feŸ (Ö2ŽŽŽŽ‘ IïcosŽ‘­~(2µG²))‘ª¨µdŽŽŽŸÒ’ܸ6²=ŽŽ’îT9(Ÿù<$‘333Ž‘33Ÿw‰feŸ (Ö2ŽŽŽŽ‘fgµ–ý¸›8à²2‘ª¨cosŽ‘¸ßµ“¸Ÿù<$‘l²1Ž‘lŸw‰feŸ (Ö4ŽŽŽŽ‘ IïsinŽ‘‘ (2µG²))˜+˜µCŽŽŽŸ¬’ܸ6²=ŽŽŸù<$’ﲇ27Ž’ï²‡Ÿw‰fe Ÿ (Ö‘€2ŽŽŽŽ’úå¼µ–ý¸›8à²18‘ª¨cosŽ‘¸ßµ“¸Ÿù<$‘l²9Ž‘lŸw‰feŸ (Ö2ŽŽŽŽ‘ IïsinŽ‘;µµ‘ñŲcosŽ‘ÿüµ“²+˜µCŽŽŽŸø‘mºUsing–«the“formulas“²3‘ª¨sinŽ‘œnµ›8&²=‘ñ µx–I§¸“²3–«ºand“²3‘ª¨cosŽ‘¸ßµ˜²=‘ñ Ÿ÷ª«¸pŽ‘ F`Ÿ÷ª«‰fe!#ŠŸUU²6µx–8ณµxŸýr±2ŽŽŽŽ‘0øºand“c‘ÿ}'om-ŽŸ ‘mbining–“çlike“terms,“this“b–ÿ}'e“c“omesŽŸ"}`Ÿö£÷Ÿù<$’‰$%²27Ž’‰$%Ÿw‰fe Ÿ (Ö‘€2ŽŽŽŽ’–sinŽ’¢I ŸûË*·±1Ž’­”²(Ÿù<$‘33µx–8ณ²3Ž‘33Ÿw‰feïûŸ (Ö‘÷ý3ŽŽŽŽ‘Va)–8ณ²6Ÿö-­«pŽ‘ Ÿö-­‰fe!#ŠŸ ÒS²6µx“¸“µxŸýr±2ŽŽŽŽ›-\k¸Ÿù<$‘l²1Ž‘lŸw‰feŸ (Ö2ŽŽŽŽ‘ ŸG(µx“¸“²3)Ÿö-­«pŽ‘ Ÿö-­‰fe!#ŠŸ ÒS²6µx“¸“µxŸýr±2ŽŽŽŽ˜²+“µCŽŽŸ¬’œb²²=Ÿù<$‘úK27Ž‘úKŸw‰fe Ÿ (Ö‘€2ŽŽŽŽ‘Ø(sinŽ‘FŸûË*·±1Ž‘'Ûº²(Ÿù<$‘33µx–8ณ²3Ž‘33Ÿw‰feïûŸ (Ö‘÷ý3ŽŽŽŽ‘Va)–8ฟù<$‘l²1Ž‘lŸw‰feŸ (Ö2ŽŽŽŽ‘ ŸG(µx“²+“9)Ÿö-­«pŽ‘ Ÿö-­‰fe!#ŠŸ ÒS²6µx“¸“µxŸýr±2ŽŽŽŽ‘-\k²+“µC(ã:ŽŽŽŽŽŸ’è²9ŽŽŒ‹ o y ý£ ? ýä‘M¸ŽŽŽ‘WxŽ‘[q˲10.6–UUFind“the“folloš¸ãwing“in˜tegrals:ŽŸdš‘Wã#5.ŽŽŽ‘mŸòc‡«ZŽŸù<$‘y÷¥²2µx–8à²+“21Ž‘yÝÞŸw‰fe!#ŠŸ (ÖµxŸýr±2Ž‘µS¸‘8à²7µxŽŽŽŽ’ßCdx².Ž©–­‘m¹Solution:ŽŽ’¢Ž*ºSinc‘ÿ}'e‘“çµxŸü^ÿ±2Ž‘µS¸›8à²7µx–Dz=“µx²(µx˜¸˜²7)–“çºwe“writeŽŸ±Ÿù<$’Ѿû²2µx–8à²+“21Ž’Ñ¥4Ÿw‰fe!#ŠŸ (ÖµxŸýr±2Ž‘µS¸‘8à²7µxŽŽŽŽ’öà ²=Ÿù<$‘úKµAŽ‘úKŸw‰fe€Ÿ (Ö‘äsxŽŽŽŽ‘æ_²+Ÿù<$‘ ØôµBŽ‘lŸw‰feïûŸ (Öx–8ณ²7ŽŽŽŽ‘Aµ;ŽŸÉ>‘mºand–O> nd“µA“ºand“µB‘ϯºby“solving“²2µx–£ž²+“21–Ç=“µA²(µx–£ž¸“²7)“+“µB‘€qxº.‘‚³Setting‘O>µx–Dz=“0ޤ ‘mºgives›)$²21–Õs=“¸²7µAº,‘Ntso˜µA“²=“¸²3º,‘Ntand˜setting˜µx“²=“7˜ºgives˜²35“=“7µB‘€qº,‘NtsoŽ¡‘mµB‘G‰²=‘Ç5º.‘™–Ther–ÿ}'efor“eŽŸ?J‘p½sŸòc‡«ZŽŸù<$‘}µ²2µx–8à²+“21Ž‘}›OŸw‰fe!#ŠŸ (ÖµxŸýr±2Ž‘µS¸‘8à²7µxŽŽŽŽ’¡œ´dx–Dz=“Ÿòc‡«ZŽŸù<$‘¤ô¸²3‘ª¨µdxŽ‘¤ôŸw‰fe]UŸ (Ö‘ ÓxŽŽŽŽ‘,n\²+‘8àŸòc‡«ZŽŸù<$‘Þ²5‘ª¨µdxŽ‘¼Ÿw‰feïûŸ (Öx–8ณ²7ŽŽŽŽ‘*=“¸²3‘ª¨lnŽ› ª§¸jµx¸j–8à²+“5‘ª¨lnŽ˜¸jµx“¸“²7¸j“²+“µC(ã:ŽŸ$<‘R㌲#12.ŽŽŽ‘mŸòc‡«ZŽŸù<$‘yÝÞ²4µxŸü^ÿ±2Ž‘µS²+–8à2µx“²+“4Ž‘yÝÞŸw‰fe7\jŸ (Ö‘ pµxŸýr±3Ž‘µS²+‘8à4µxŽŽŽŽ’´#dx².ަ‘m¹Solution:ŽŽ’¢Ž*ºSinc‘ÿ}'e‘“çµxŸü^ÿ±3Ž›µS²+‘8à4µx–Dz=“µx²(µxŸü^ÿ±2Ž˜²+‘8à4)º,–“çwe“writeŽŸÿ•Ÿù<$’Ârв4µxŸü^ÿ±2Ž‘µS²+–8à2µx“²+“4Ž’ÂrПw‰fe7\jŸ (Ö‘ pµxŸýr±3Ž‘µS²+‘8à4µxŽŽŽŽ’ýÉ…²=Ÿù<$‘úKµAx–8à²+“µBŽ‘úKŸw‰fe!†4Ÿ (Ö‘ ãxŸýr±2Ž‘µS²+‘8à4ŽŽŽŽ‘(ì’+Ÿù<$‘lµCŽ‘lŸw‰feÜÌŸ (Ö‘ØxŽŽŽŽŽŸ;‘mºand–“ç nd“µAº,“µB‘Xºand“µC‘Kºby“solving:ޤ—à‘nͲ4µxŸûÞÿ±2Ž‘µS²+–8à2µx“²+“4ŽŽ’°! =ŽŽ’Áè>(µAx–8à²+“µB‘€q²)µx“²+“µC‘·²(µxŸûÞÿ±2Ž‘µS²+“4)ŽŽŽŸ’°! =ŽŽ’Áè>µAxŸûÞÿ±2Ž›µS²+–8àµB‘€qx“²+“µC‘·xŸûÞÿ±2Ž˜²+“4µC‘~4²=‘Ç(µA“²+“µC‘·²)µxŸûÞÿ±2Ž˜²+“µB‘€qx“²+“4µC(ã:ŽŽŽ¡‘mºEquating›“çc–ÿ}'o“ecients˜of˜²1º,˜µx˜ºand˜µxŸü^ÿ±2Ž‘ZºgivesŽ¡’¾-µA–8à²+“µC‘~4²=–Ç4;› ㈵B‘G‰²=“2;˜4µC‘~4²=“4µ;Ž¡‘mºso–“çµC‘~4²=›Ç1º,“µB‘G‰²=˜2“ºand“µA˜²=˜3º.‘™–Ther–ÿ}'efor“eޤq\‘}OžŸòc‡«ZŽŸù<$’Š-z²4µxŸü^ÿ±2Ž‘µS²+–8à2µx“²+“4Ž’Š-zŸw‰fe7\jŸ (Ö‘ pµxŸýr±3Ž‘µS²+‘8à4µxŽŽŽŽ’Äg¿dxŽŽ’ÙSM²=ŽŽ’ëkŸòc‡«ZŽŸù<$’úÕ²3µx‘ª¨dxŽ’÷øGŸw‰felnŸ (ÖxŸýr±2Ž‘µS²+‘8à4ŽŽŽŽ’ÐÈ+‘8àŸòc‡«ZŽŸù<$‘ز2‘ª¨µdxŽ‘¼Ÿw‰felnŸ (ÖxŸýr±2Ž‘µS²+‘8à4ŽŽŽŽ‘-ï=+‘8àŸòc‡«ZŽŸù<$‘¼µdxŽ‘¼Ÿw‰fe 뎟 (Ö‘š9xŽŽŽŽŽŽŽŸÒ’ÙSM²=ŽŽŸù<$’ìMž3Ž’ìMžŸw‰feŸ (Ö2ŽŽŽŽ’ô+zlnŽ’ü€Ñ(µxŸûÞÿ±2Ž‘µS²+–8à4)“+“tanŽ‘ªªŸûÞÿ·±1ŽŸù<$‘DùµxŽ‘DùŸw‰fe·Ÿ (Ö[޲2ŽŽŽŽ‘'h(+“lnŽ‘ 8߸jµx¸j“²+“µC(ã;ŽŽŽŸ—‘mºwher›ÿ}'e–“çthe“ rst“inte˜gr˜al“was“found“using“the“substitution“µu–Dz=“µxŸü^ÿ±2Ž‘µS²+‘8à4º.ŽŸ¯Ö‘R㌲#14.ŽŽŽ‘mŸòc‡«ZŽŸù<$‘|/3µxŸü^ÿ±4Ž‘'µdxŽ‘yÝÞŸw‰felnŸ (ÖxŸýr±2Ž‘µS²+‘8à4ŽŽŽŽ’–}.ަ‘m¹Solution:ŽŽ’¢Ž*ºL‘ÿ}'ong–¹¬division“of“µxŸü^ÿ±4Ž‘6ºby“µxŸü^ÿ±2Ž›б²+‘T>4“ºgives“a“quotient“of“µxŸü^ÿ±2Ž˜¸‘T>²4ŽŸ ‘mºwith–“çr‘ÿ}'emainder“²16º,“soŽ¡‘n„vŸòc‡«ZŽŸù<$‘}³§µxŸü^ÿ±4Ž‘'µdxŽ‘{bRŸw‰felnŸ (ÖxŸýr±2Ž‘µS²+‘8à4ŽŽŽŽ’šÉ =‘ÇŸòc‡«ZŽ‘qÁŸñæ^Ž‘Î5µxŸûÞÿ±2Ž›µS¸–8à²4“+Ÿù<$‘ "I16Ž‘lŸw‰felnŸ (ÖµxŸýr±2Ž˜²+“4ŽŽŽŽ‘ ´Ÿñæ^«ŽŽ‘eøµdx‘Dz=Ÿù<$‘úK1Ž‘úKŸw‰feŸ (Ö3ŽŽŽŽ‘ -µxŸûÞÿ±3Ž‘µS¸–8à²4µx“²+“8‘ª¨tanŽ‘rŸûÞÿ·±1ŽŸù<$‘¶ÁµxŽ‘¶ÁŸw‰fe·Ÿ (Ö[޲2ŽŽŽŽ‘&Ùð+“µC(ã:ŽŽŸ’åÿ²10ŽŽŒ‹ F y ý£ ? ýçœy‘R㌲#24.ŽŽŽ‘mŸòc‡«ZŽŸù<$’„ÒÀ²1Ž‘yÝÞŸw‰feéÄŸ (Ö1–8à+“µeŸýr´xŽŽŽŽŽ’—¥}µdx².ŽŸ)‘m¹Solution:ŽŽ’¢Ž*ºFirst–“çmake“the“substitution“µu–Dz=“µeŸü^ÿ´xŽ‘äº,›“çµdu“²=“µeŸü^ÿ´xŽ‘³Œµdxº,˜soޤv.’ªÝŸòc‡«ZŽŸù<$’Â`›²1Ž’·k¹Ÿw‰feéÄŸ (Ö1–8à+“µeŸýr´xŽŽŽŽŽ’Õ3Xµdx–Dz=“Ÿòc‡«ZŽŸù<$‘LôµeŸü^ÿ´xŽ‘³ŒµdxŽ‘¤ôŸw‰fe#—Ÿ (ÖeŸýr±2´xŽ‘ >7²+‘8àµeŸýr´xŽŽŽŽŽ‘76Z²=“Ÿòc‡«ZŽŸù<$‘B#µduŽ‘¤ôŸw‰fe(MŸ (ÖuŸýr±2Ž‘µS²+‘8àµuŽŽŽŽ‘-t:ŽŸ‘mºNow‘“çµuŸü^ÿ±2Ž‘µS²+›8àµu–Dz=“µu²(µu˜²+˜1)º,–“çso“writeŽŸçôŸù<$’ߵDz1Ž’Ô!¡Ÿw‰fe(MŸ (ÖµuŸýr±2Ž‘µS²+‘8àµuŽŽŽŽ’ôD9²=Ÿù<$‘úKµAŽ‘úKŸw‰fe€Ÿ (Ö‘ãBuŽŽŽŽ‘æ_²+Ÿù<$‘ Ú%µBŽ‘lŸw‰feò]Ÿ (Öu–8à²+“1ŽŽŽŽ‘‘£µ;ŽŸ"‘mºand–ro nd“µA“ºand“µB‘òàºby“solving“²1–Ç=“µA²(µu–ð²+“1)“+“µB‘€quº.‘ŽnSetting›roµu–Dz=“0˜ºgivesŽ© ‘m²1–Ç=“µAº,–“çand“setting“µu–Dz=“¸²1–“çºgives“²1–Ç=“¸µB‘€qº,–“çso“µB‘G‰²=‘Ǹ²1º.‘™–Ther–ÿ}'efor“eŽ¡’¬n·Ÿòc‡«ZŽŸù<$’ÁéµduŽ’¹L“Ÿw‰fe(MŸ (ÖuŸýr±2Ž‘µS²+‘8àµuŽŽŽŽŽŽ’ਲ=ŽŽ’òo1Ÿòc‡«ZŽŸù<$’ÿM µduŽ’ÿM Ÿw‰fe íðŸ (Ö‘š9uŽŽŽŽ’ §¸‘8àŸòc‡«ZŽŸù<$‘óµduŽ‘¼Ÿw‰feò]Ÿ (Öu–8à²+“1ŽŽŽŽŽŽŽŸz’à¨=ŽŽ’òo1lnŽ’üo0¸jµu¸j–8à“²lnŽ‘ 8߸jµu“²+“1¸j“²+“µCŽŽŽ¤’ਲ=ŽŽ’òo1lnŽ’úĈ(µeŸûÞÿ´xŽ‘ä²)–8ณ²lnŽ‘ Ž7(µeŸûÞÿ´xŽ‘AIJ+“1)“+“µCŽŽŽ¡’ਲ=ŽŽ’òo1µx–8ณ²lnŽ‘ Ž7(µeŸûÞÿ´xŽ‘AIJ+“1)“+“µC(ã:ŽŽŽŸ‘¸‘W¸xŽ‘[q˲10.7–UUFind“the“folloš¸ãwing“in˜tegrals“b˜y“the“methoGd“of“in˜tegration“b˜y“parts:ŽŸ(‘Wã#4.ŽŽŽ‘mŸòc‡«ZŽ‘xª«µxeŸûÞÿ´axŽ‘ üµdx².ŽŸ0ž‘m¹Solution:ŽŽ’¢Ž*ºL‘ÿ}'et›ÎÕµu–1Ù²=“µxº,‘ݵdv‘²²=“µeŸü^ÿ´axŽ‘ üµdxº.‘J_Then˜µdu“²=“µdx˜ºand˜µv›²²=“Ÿ÷ñÄ«RŽ‘ ‡,µdv˜²=ŽŸCàŸù<$‘nWÍ1Ž‘n35Ÿw‰feI0Ÿ (ÖµaŽŽŽŽ‘t¯˜eŸûÞÿ´axŽ‘ _Tº,‘“çsoŽŸGF’†ñ÷Ÿòc‡«ZŽ’’œ µxeŸûÞÿ´axŽ– üµdx›Ç²=Ÿù<$‘ã1Ž‘úKŸw‰feI0Ÿ (ÖµaŽŽŽŽ‘ v®xeŸûÞÿ´axŽ‘ ˜4¸Ÿù<$‘«²1Ž‘lŸw‰feI0Ÿ (ÖµaŽŽŽŽ‘ “Ÿòc‡«ZŽ‘=ǵeŸûÞÿ´axŽ“µdx˜²=Ÿù<$‘ã1Ž‘úKŸw‰feI0Ÿ (ÖµaŽŽŽŽ‘ v®xeŸûÞÿ´axŽ– ˜4¸Ÿù<$‘Îä²1Ž‘lŸw‰fe Å£Ÿ (ÖµaŸýr±2ŽŽŽŽŽ‘déµeŸûÞÿ´axŽ“²+‘8àµC(ã:ŽŸ!Ê«‘Wã²#8.ŽŽŽ‘mŸòc‡«ZŽ‘xª«²sinŽ’„ñÉŸûË*·±1Ž’‘Xåµx‘ª¨dx².ŽŸc‘m¹Solution:ŽŽ’¢Ž*ºL‘ÿ}'et›6Öµu–îB²=“sinŽ‘5`ŸûË*·±1Ž‘œ|µxº,‘_‘µdv‘J²=“µdxº.‘‚bThen˜µdu“²=“(1–®ñ¸“µxŸü^ÿ±2Ž‘|s²)Ÿü^ÿ·±1´=±2Ž‘|sµdxަ‘mºand–“çµv‘"ñ²=‘ǵxº,“soޤ]’›  Ÿòc‡«ZŽ’§Jµ²sinŽ’³‘ÓŸûË*·±1Ž’¿øïµx›ª¨dx–Dz=“µx˜²sinŽ‘ ñÆŸûË*·±1Ž‘Xâµx–8ณŸòc‡«ZŽ‘ ㉵x²(1“¸“µxŸûÞÿ±2Ž‘|s²)ŸûÞÿ·±1´=±2Ž‘|sµdx:ŽŸ‘mºThe–“çsubstitution“µw› û²=‘Ç1–8ณµxŸü^ÿ±2Ž‘|sº,–“çµdw˜²=‘Ǹ²2µx‘ª¨dx“ºgivesŽ¡‘mŸòc‡«ZŽ‘xª«µx²(1–ø“µxŸûÞÿ±2Ž‘|s²)ŸûÞÿ·±1´=±2Ž‘|sµdx–Dz=“¸Ÿù<$‘33²1Ž‘33Ÿw‰feŸ (Ö2ŽŽŽŽ‘ Ÿòc‡«ZŽ‘»¸µwDãŸûÞÿ·±1´=±2Ž‘®µdw‘ û²=“¸µwDãŸûÞÿ±1´=±2Ž‘ š<²+›ÃµC‘~4²=“¸Ÿö-­«pŽ‘ Ÿö-­‰felnŸ ÒS²1–8ณµxŸýr±2ŽŽŽŽ‘&/þ²+˜µC(ã:ŽŸ]‘mºTher–ÿ}'efor“eŽ¡’©$ÅŸòc‡«ZŽ’´Ïn²sinŽ’ÁŒŸûË*·±1Ž’Í}¨µx›ª¨dx–Dz=“µx˜²sinŽ‘ ñÆŸûË*·±1Ž‘Xâµx–8à²+“Ÿö-­«pŽ‘ 8áŸö-­‰felnŸ ÒS²1“¸“µxŸýr±2ŽŽŽŽ‘)Þ/²+“µC(ã:ŽŽŸ’åÿ²11ŽŽŒ‹ Œò y ý£ ? ýçœy‘R㌲#13.ŽŽŽ‘mŸòc‡«ZŽ‘xª«µx‘ª¨²secŽ‘ýŸûÞÿ±2Ž‘§µx‘ª¨dx².ŽŸÀS‘m¹Solution:ŽŽ’¢Ž*ºL‘ÿ}'et›ÞÔµu–•²=“µxº,‘1µdv‘zn²=“secŽ‘óêŸü^ÿ±2Ž‘µx‘ª¨dxº.‘z\Then˜µdu“²=“µdx˜ºand˜µv‘zn²=ޤ ‘mŸ÷ñÄ«RŽ‘uUUµdv‘"ñ²=‘ÇtanŽ‘㊵xº,‘“çsoŽŸª¶‘z'ñŸòc‡«ZŽ’…Òšµx‘ª¨²secŽ‘ýŸûÞÿ±2Ž‘§µx›ª¨dx–Dz=“µx˜²tanŽ‘Çµx–8ณŸòc‡«ZŽ‘ ㉲tanŽ‘ÿûµx˜dx–Dz=“µx˜²tanŽ‘Çµx–8ณ²lnŽ‘ 8߸j˜²secŽ‘*¥µx¸j“²+“µC(ã:ŽŸ#8ó‘R㌲#18.ŽŽŽ‘mŸòc‡«ZŽ‘w²(lnŽ‘ ÿÿµx²)ŸûÞÿ±2Ž‘'µdx².ŽŸR‘m¹Solution:ŽŽ’¢Ž*ºL‘ÿ}'et›ÿCµu–YX²=“(lnŽ‘ ÿÿµx²)Ÿü^ÿ±2Ž‘|sº,‘Zµdv‘µ1²=“µdxº.‘Û«Then˜µdu“²=Ÿù<$‘Œ‹2‘ª¨lnŽ‘ ª§µxŽ‘Œ‹Ÿw‰feaÄŸ (Ö‘UTxŽŽŽŽ‘!‚dx˜ºandŽ¡‘mµv‘"ñ²=‘ǵxº,‘“çsoŽŸŽ@’²b¶Ÿòc‡«ZŽ’¼b·²(lnŽ› ÿÿµx²)ŸûÞÿ±2Ž‘'µdx–Dz=“µx²(lnŽ˜µx²)ŸûÞÿ±2Ž‘µS¸‘8à²2‘ª¨Ÿòc‡«ZŽ‘ UQ²lnŽ‘UPµx‘ª¨dx:ŽŸª¶‘mºT‘ÿ;¼o–=à nd“Ÿ÷ñÄ«RŽ‘ “²lnŽ‘“µx‘ª¨dxº,›h:we“use“inte–ÿ}'gr“ation–=Ãby“p‘ÿ}'arts“again,˜with“µu–úβ=“lnŽ‘ ú͵xº,Ž¡‘mµdv‘"ñ²=‘ǵdxº,‘“çgivingŽŸŽ@’¦káŸòc‡«ZŽ’°kâ²(lnŽ› ÿÿµx²)ŸûÞÿ±2Ž‘'µdx–Dz=“µx²(lnŽ˜µx²)ŸûÞÿ±2Ž‘µS¸–8à²2µx‘ª¨²lnŽ‘ ª§µx“²+“2µx“²+“µC(ã:ŽŸŽA‘M¸ŽŽŽ‘WxŽ‘[q˲10.7‘UU#24.ŽŸ‘WThe–ìregion“under“the“curvš¸ãe“µy‘"ñ²=‘ÇsinŽ‘¸Þµx“²bGet˜w˜een“µx–Dz=“0–ìand“µx–Dz=“µ‘Gñ²is‘ìrev˜olv˜edŽ¡‘WabGout–»ÿthe“µy[Ù²-axis.‘¥ÆFind“the“vš¸ãolume“of“the“resulting“solid“(a)“b˜y“the“shellŽ¡‘WmethoGd;–UUand“(b)“bš¸ãy“the“w˜asher“methoGd.ŽŸ§%‘W¹Solution:ŽŽ’ŒŽ*º(a)–½õUsing“shel‘‚Øls,–ÈyµdV›L+²=‘G2µ[Ùx‘ª¨²sinŽ‘œnµx‘ª¨dxº,“so‘½õµV˜²=‘GŸòc‡«ZŽŸôÜi‘ H´ŽŸ@‘¡±0ŽŽ‘+_²2µ[Ùx‘ª¨²sinŽ‘œnµx‘ª¨dxº.ŽŸ*µ‘WL‘ÿ}'et› Kµu–¢ÿ²=“µx˜ºand˜µdv‘þز=“sinŽ‘”ŵx‘ª¨dxº,‘+£so˜µdu“²=“µdx˜ºand˜µv›þز=“Ÿ÷ñÄ«RŽ‘ øRµdv˜²=“¸‘ª¨²cosŽ‘¸ßµxº.‘ÁThisŽ¡‘WgivesŽŸqΑw»µVŽ’‰É?²=ŽŽ’›]2µ‘Ÿòc‡«ZŽŸôÜi‘ ‚´ŽŸ@‘”»±0ŽŽ‘™µx‘ª¨²sinŽ‘œnµx›ª¨dx–Dz=“2µ‘Ÿñæ^«Ž‘ bõ¸µx˜²cosŽ‘¸ßµx²]ŸûÞÿ´ŽŸ™á±0ŽŽ‘¦O²+‘8àŸòc‡«ZŽŸôÜi‘ 8á´ŽŸ@‘DZ0ŽŽ‘Pø²cosŽ‘"_/µx˜dxŸñæ^«ŽŽŽŽŽŸ#á’‰É?²=ŽŽ’›]2µ‘9´²(¸µx‘ª¨²cosŽ‘¸ßµx–8à²+“sinŽ‘*¦µx²)]ŽŽŸúøâ‘Q—ä´ŽŸ‘Q—ä±0ŽŽ‘YÌk²=‘Ç2µ[ÙŸûÞÿ±2Ž‘ØLµ:ŽŽŽ©‘Wº(b)–§The“horizontal“cr–ÿ}'oss-se“ctions›§ar“e˜washers˜with˜inner˜r“adius˜µx–Dz=“sinŽ‘6ŸûË*·±1Ž‘uRµyŽ¡‘Wºand–“çouter“r›ÿ}'adius“µx–Dz=“µ‘”¹¸‘8à²sinŽ‘þŸûË*·±1Ž‘çµy[Ùº.‘™–Ther˜efor˜eަ‘nµdV‘ÿü²=›Çµ•[Ù²[(µ‘”¹¸‘8à²sinŽ‘þŸûË*·±1Ž‘çµy“²)ŸûÞÿ±2Ž‘µS¸‘8à²(sinŽ‘ GŸûË*·±1Ž‘®:µy“²)ŸûÞÿ±2Ž‘|s²]‘ª¨µdy‘"ñ²=˜µ“²(µ“ŸûÞÿ±2Ž‘,¸‘8à²2µ‘²sinŽ‘MŸŸûË*·±1Ž‘´»µy“²)‘ª¨µdy“;ަ‘Wºwhich‘“çgivesŽŸ˜6‘~üĵV‘ÿü²=‘ÇŸòc‡«ZŽŸôÜi‘ DZ1ŽŸ@‘UR0ŽŽ‘CŒ²(µ[ÙŸûÞÿ±3Ž›,¸‘8à²2µ[ÙŸûÞÿ±2Ž‘‚ô²sinŽ‘ÊŸûË*·±1Ž‘1.µy•[Ù²)‘ª¨µdy‘"ñ²=‘ǵ“ŸûÞÿ±3Ž˜¸‘8à²2µ“ŸûÞÿ±2Ž‘‚ôŸòc‡«ZŽŸôÜi‘‚õ±1ŽŸ@‘ .0ŽŽ‘ª²sinŽ‘"ñ.ŸûË*·±1Ž‘/XJµy‘dy“:ŽŽŸ’åÿ²12ŽŽŒ‹ ™Ž y ý£ ? ýä‘WºT‘ÿ;¼o–Çevaluate“the“last“inte–ÿ}'gr“al,›âlet‘ǵu–ó“²=“sinŽ‘:±ŸûË*·±1Ž‘¡Íµy[Ùº,˜µdv‘Ol²=“µdy[Ùº,˜so‘ǵdu“²=“(1‘x¸Ž© ‘Wµy[ÙŸü^ÿ±2Ž‘ØL²)Ÿü^ÿ·±1´=±2Ž‘Ñ˵dy‘ïÀºand–“çµv‘"ñ²=‘ǵy[Ùº.‘™–This“givesŽŸ&o’‚÷•Ÿòc‡«ZŽŸôÜi’Œ÷–±1ŽŸ@’ˆ…Ï0ŽŽ’“±²sinŽ’ŸeÏŸûË*·±1Ž’«Ìëµy‘dyŽŽ’Ç1õ²=ŽŽ’Ú,Fµy‘²sinŽ‘MŸŸûË*·±1Ž‘´»µy[ÙŸ÷æb«ŽŽŸùÁ4’5ı1ŽŸ >Ï’5Ä0ŽŽ’ 븑8àŸòc‡«ZŽŸôÜi‘ 8á±1ŽŸ@‘Ç0ŽŽ‘_üµyš[Ù²(1–8ณµy˜ŸûÞÿ±2Ž‘ØL²)ŸûÞÿ·±1´=±2Ž‘|sµdyŽŽŽŸÂè’Ç1õ²=ŽŽŸù<$’Ú,FµŽ’Ú,FŸw‰feŸ (Ö‘‡²2ŽŽŽŽ’ã§w+‘8àŸòc‡«ZŽŸôÜi‘ 8á±0ŽŸ@‘Ç1ŽŽŸù<$‘“/²1Ž‘“/Ÿw‰feŸ (Ö2ŽŽŽŽ‘ÆcµwDãŸûÞÿ·±1´=±2Ž‘ÁVµdwŽŽŽŸ¦t’Ç1õ²=ŽŽŸù<$’Ú,FµŽ’Ú,FŸw‰feŸ (Ö‘‡²2ŽŽŽŽ’ã§w+‘lŸôæ`«Ž‘ d÷µwDãŸûÞÿ±1´=±2Ž‘ Ö­Ÿôæ`«ŽŽ‘#]GŸôæ`iŽŽŸöøÝ‘(+±0ŽŸ(‘(+1ŽŽŽŽŽŸ’Ç1õ²=ŽŽŸù<$’Ú,FµŽ’Ú,FŸw‰feŸ (Ö‘‡²2ŽŽŽŽ’ã§w¸‘8à²1ŽŽŽ¤ÿâ‘Wº(we–“çsubstitute›ÿ}'d“µw‘ û²=‘Ç1–8ณµy[ÙŸü^ÿ±2Ž‘ØLº).‘™–Ther˜efor˜eŽŸ’’¸—~µV‘ÿü²=›Çµ[ÙŸûÞÿ±3Ž‘,¸–8à²2µ[ÙŸûÞÿ±2Ž‘ØL²(Ÿù<$‘33µŽ‘33Ÿw‰feŸ (Ö‘‡²2ŽŽŽŽ‘ ®d¸“²1)˜=˜2µ[ÙŸûÞÿ±2Ž‘ØLµ:Ž¡‘WºA¾“lternatively,‘G"we–ðc›ÿ}'an“c˜onvert“this“inte˜gr˜and“µdV‘v¾²=‘=Úµ[Ù²[(µ‘‘¸–5.µx²)Ÿü^ÿ±2Ž‘±¡¸“µxŸü^ÿ±2Ž‘|s²]‘ª¨µdyަ‘Wºentir›ÿ}'ely–“çto“µxº.‘™–Sinc˜e“µy›"ñ²=‘ÇsinŽ‘¸Þµx“ºthen“µdy˜²=‘ÇcosŽ‘ÕOµx‘ª¨dxº.‘™–Thus“our“inte–ÿ}'gr“al‘“çisŽŸU^’¾Ã÷Ÿòc‡«ZŽŸôÜi’ÈÃø´@L=±2ŽŸ@’ÄR10ŽŽ’×ñfµ•[Ù²(µ“ŸûÞÿ±2Ž‘,¸‘8à²2µ“x²)‘ª¨sinŽ‘œnµx‘ª¨dxŽŸ@‘WºIf–“çyou“che‘ÿ}'ck,“this“gives“the“same“answer.ŽŽŸ’åÿ²13ŽŽŒø¥9ƒ’À;èy— óý': cmti10óò"V cmbx10ó !",š cmsy10ó O!â…cmsy7ó °Ü0ncmsy5ó  b> cmmi10ó 0e—rcmmi7óKñ`y cmr10óÙ“ Rcmr7ó†›Zcmr5óú±u cmex10ù«íßßßßßß