反常积分技巧篇

设.下述两个等式成立：

%5E2%20%5Cright)%5Cmathrm%7Bd%7D%20x%20%26amp%3B%3D%20%5Cfrac%7B1%7D%7Ba%7D%5Cint_0%5E%5Cinfty%20f(x%5E2)%5Cmathrm%7Bd%7D%20x%2C%20%5C%5C%0A%5Cint_0%5E%5Cinfty%20f%5Cleft(ax%2B%20%5Cfrac%7Bb%7D%7Bx%7D%5Cright)%20%5Cmathrm%7Bd%7D%20x%20%26amp%3B%3D%20%5Cfrac%7B1%7D%7Ba%7D%20%5Cint_0%5E%7B%5Cinfty%7Df(%5Csqrt%7Bx%5E2%2B4ab%7D)%20%5Cmathrm%7Bd%7D%20x.%0A%5Cend%7Balign%7D%3C%2Ftitle%3E%0A%3Cdefs%20aria-hidden%3D%22true%22%3E%0A%3Cpath%20stroke-width%3D%221%22%20id%3D%22E1-MJSZ2-222B%22%20d%3D%22M114%20-798Q132%20-824%20165%20-824H167Q195%20-824%20223%20-764T275%20-600T320%20-391T362%20-164Q365%20-143%20367%20-133Q439%20292%20523%20655T645%201127Q651%201145%20655%201157T672%201201T699%201257T733%201306T777%201346T828%201360Q884%201360%20912%201325T944%201245Q944%201220%20932%201205T909%201186T887%201183Q866%201183%20849%201198T832%201239Q832%201287%20885%201296L882%201300Q879%201303%20874%201307T866%201313Q851%201323%20833%201323Q819%201323%20807%201311T775%201255T736%201139T689%20936T633%20628Q574%20293%20510%20-5T410%20-437T355%20-629Q278%20-862%20165%20-862Q125%20-862%2092%20-831T55%20-746Q55%20-711%2074%20-698T112%20-685Q133%20-685%20150%20-700T167%20-741Q167%20-789%20114%20-798Z%22%3E%3C%2Fpath%3E%0A%3Cpath%20stroke-width%3D%221%22%20id%3D%22E1-MJMAIN-221E%22%20d%3D%22M55%20217Q55%20305%20111%20373T254%20442Q342%20442%20419%20381Q457%20350%20493%20303L507%20284L514%20294Q618%20442%20747%20442Q833%20442%20888%20374T944%20214Q944%20128%20889%2059T743%20-11Q657%20-11%20580%2050Q542%2081%20506%20128L492%20147L485%20137Q381%20-11%20252%20-11Q166%20-11%20111%2057T55%20217ZM907%20217Q907%20285%20869%20341T761%20397Q740%20397%20720%20392T682%20378T648%20359T619%20335T594%20310T574%20285T559%20263T548%20246L543%20238L574%20198Q605%20158%20622%20138T664%2094T714%2061T765%2051Q827%2051%20867%20100T907%20217ZM92%20214Q92%20145%20131%2089T239%2033Q357%2033%20456%20193L425%20233Q364%20312%20334%20337Q285%20380%20233%20380Q171%20380%20132%20331T92%20214Z%22%3E%3C%2Fpath%3E%0A%3Cpath%20stroke-width%3D%221%22%20id%3D%22E1-MJMAIN-30%22%20d%3D%22M96%20585Q152%20666%20249%20666Q297%20666%20345%20640T423%20548Q460%20465%20460%20320Q460%20165%20417%2083Q397%2041%20362%2016T301%20-15T250%20-22Q224%20-22%20198%20-16T137%2016T82%2083Q39%20165%2039%20320Q39%20494%2096%20585ZM321%20597Q291%20629%20250%20629Q208%20629%20178%20597Q153%20571%20145%20525T137%20333Q137%20175%20145%20125T181%2046Q209%2016%20250%2016Q290%2016%20318%2046Q347%2076%20354%20130T362%20333Q362%20478%20354%20524T321%20597Z%22%3E%3C%2Fpath%3E%0A%3Cpath%20stroke-width%3D%221%22%20id%3D%22E1-MJMATHI-66%22%20d%3D%22M118%20-162Q120%20-162%20124%20-164T135%20-167T147%20-168Q160%20-168%20171%20-155T187%20-126Q197%20-99%20221%2027T267%20267T289%20382V385H242Q195%20385%20192%20387Q188%20390%20188%20397L195%20425Q197%20430%20203%20430T250%20431Q298%20431%20298%20432Q298%20434%20307%20482T319%20540Q356%20705%20465%20705Q502%20703%20526%20683T550%20630Q550%20594%20529%20578T487%20561Q443%20561%20443%20603Q443%20622%20454%20636T478%20657L487%20662Q471%20668%20457%20668Q445%20668%20434%20658T419%20630Q412%20601%20403%20552T387%20469T380%20433Q380%20431%20435%20431Q480%20431%20487%20430T498%20424Q499%20420%20496%20407T491%20391Q489%20386%20482%20386T428%20385H372L349%20263Q301%2015%20282%20-47Q255%20-132%20212%20-173Q175%20-205%20139%20-205Q107%20-205%2081%20-186T55%20-132Q55%20-95%2076%20-78T118%20-61Q162%20-61%20162%20-103Q162%20-122%20151%20-136T127%20-157L118%20-162Z%22%3E%3C%2Fpath%3E%0A%3Cpath%20stroke-width%3D%221%22%20id%3D%22E1-MJMAIN-28%22%20d%3D%22M94%20250Q94%20319%20104%20381T127%20488T164%20576T202%20643T244%20695T277%20729T302%20750H315H319Q333%20750%20333%20741Q333%20738%20316%20720T275%20667T226%20581T184%20443T167%20250T184%2058T225%20-81T274%20-167T316%20-220T333%20-241Q333%20-250%20318%20-250H315H302L274%20-226Q180%20-141%20137%20-14T94%20250Z%22%3E%3C%2Fpath%3E%0A%3Cpath%20stroke-width%3D%221%22%20id%3D%22E1-MJMATHI-61%22%20d%3D%22M33%20157Q33%20258%20109%20349T280%20441Q331%20441%20370%20392Q386%20422%20416%20422Q429%20422%20439%20414T449%20394Q449%20381%20412%20234T374%2068Q374%2043%20381%2035T402%2026Q411%2027%20422%2035Q443%2055%20463%20131Q469%20151%20473%20152Q475%20153%20483%20153H487Q506%20153%20506%20144Q506%20138%20501%20117T481%2063T449%2013Q436%200%20417%20-8Q409%20-10%20393%20-10Q359%20-10%20336%205T306%2036L300%2051Q299%2052%20296%2050Q294%2048%20292%2046Q233%20-10%20172%20-10Q117%20-10%2075%2030T33%20157ZM351%20328Q351%20334%20346%20350T323%20385T277%20405Q242%20405%20210%20374T160%20293Q131%20214%20119%20129Q119%20126%20119%20118T118%20106Q118%2061%20136%2044T179%2026Q217%2026%20254%2059T298%20110Q300%20114%20325%20217T351%20328Z%22%3E%3C%2Fpath%3E%0A%3Cpath%20stroke-width%3D%221%22%20id%3D%22E1-MJMATHI-78%22%20d%3D%22M52%20289Q59%20331%20106%20386T222%20442Q257%20442%20286%20424T329%20379Q371%20442%20430%20442Q467%20442%20494%20420T522%20361Q522%20332%20508%20314T481%20292T458%20288Q439%20288%20427%20299T415%20328Q415%20374%20465%20391Q454%20404%20425%20404Q412%20404%20406%20402Q368%20386%20350%20336Q290%20115%20290%2078Q290%2050%20306%2038T341%2026Q378%2026%20414%2059T463%20140Q466%20150%20469%20151T485%20153H489Q504%20153%20504%20145Q504%20144%20502%20134Q486%2077%20440%2033T333%20-11Q263%20-11%20227%2052Q186%20-10%20133%20-10H127Q78%20-10%2057%2016T35%2071Q35%20103%2054%20123T99%20143Q142%20143%20142%20101Q142%2081%20130%2066T107%2046T94%2041L91%2040Q91%2039%2097%2036T113%2029T132%2026Q168%2026%20194%2071Q203%2087%20217%20139T245%20247T261%20313Q266%20340%20266%20352Q266%20380%20251%20392T217%20404Q177%20404%20142%20372T93%20290Q91%20281%2088%20280T72%20278H58Q52%20284%2052%20289Z%22%3E%3C%2Fpath%3E%0A%3Cpath%20stroke-width%3D%221%22%20id%3D%22E1-MJMAIN-2212%22%20d%3D%22M84%20237T84%20250T98%20270H679Q694%20262%20694%20250T679%20230H98Q84%20237%2084%20250Z%22%3E%3C%2Fpath%3E%0A%3Cpath%20stroke-width%3D%221%22%20id%3D%22E1-MJMATHI-62%22%20d%3D%22M73%20647Q73%20657%2077%20670T89%20683Q90%20683%20161%20688T234%20694Q246%20694%20246%20685T212%20542Q204%20508%20195%20472T180%20418L176%20399Q176%20396%20182%20402Q231%20442%20283%20442Q345%20442%20383%20396T422%20280Q422%20169%20343%2079T173%20-11Q123%20-11%2082%2027T40%20150V159Q40%20180%2048%20217T97%20414Q147%20611%20147%20623T109%20637Q104%20637%20101%20637H96Q86%20637%2083%20637T76%20640T73%20647ZM336%20325V331Q336%20405%20275%20405Q258%20405%20240%20397T207%20376T181%20352T163%20330L157%20322L136%20236Q114%20150%20114%20114Q114%2066%20138%2042Q154%2026%20178%2026Q211%2026%20245%2058Q270%2081%20285%20114T318%20219Q336%20291%20336%20325Z%22%3E%3C%2Fpath%3E%0A%3Cpath%20stroke-width%3D%221%22%20id%3D%22E1-MJMAIN-29%22%20d%3D%22M60%20749L64%20750Q69%20750%2074%20750H86L114%20726Q208%20641%20251%20514T294%20250Q294%20182%20284%20119T261%2012T224%20-76T186%20-143T145%20-194T113%20-227T90%20-246Q87%20-249%2086%20-250H74Q66%20-250%2063%20-250T58%20-247T55%20-238Q56%20-237%2066%20-225Q221%20-64%20221%20250T66%20725Q56%20737%2055%20738Q55%20746%2060%20749Z%22%3E%3C%2Fpath%3E%0A%3Cpath%20stroke-width%3D%221%22%20id%3D%22E1-MJMAIN-32%22%20d%3D%22M109%20429Q82%20429%2066%20447T50%20491Q50%20562%20103%20614T235%20666Q326%20666%20387%20610T449%20465Q449%20422%20429%20383T381%20315T301%20241Q265%20210%20201%20149L142%2093L218%2092Q375%2092%20385%2097Q392%2099%20409%20186V189H449V186Q448%20183%20436%2095T421%203V0H50V19V31Q50%2038%2056%2046T86%2081Q115%20113%20136%20137Q145%20147%20170%20174T204%20211T233%20244T261%20278T284%20308T305%20340T320%20369T333%20401T340%20431T343%20464Q343%20527%20309%20573T212%20619Q179%20619%20154%20602T119%20569T109%20550Q109%20549%20114%20549Q132%20549%20151%20535T170%20489Q170%20464%20154%20447T109%20429Z%22%3E%3C%2Fpath%3E%0A%3Cpath%20stroke-width%3D%221%22%20id%3D%22E1-MJSZ3-28%22%20d%3D%22M701%20-940Q701%20-943%20695%20-949H664Q662%20-947%20636%20-922T591%20-879T537%20-818T475%20-737T412%20-636T350%20-511T295%20-362T250%20-186T221%2017T209%20251Q209%20962%20573%201361Q596%201386%20616%201405T649%201437T664%201450H695Q701%201444%20701%201441Q701%201436%20681%201415T629%201356T557%201261T476%201118T400%20927T340%20675T308%20359Q306%20321%20306%20250Q306%20-139%20400%20-430T690%20-924Q701%20-936%20701%20-940Z%22%3E%3C%2Fpath%3E%0A%3Cpath%20stroke-width%3D%221%22%20id%3D%22E1-MJSZ3-29%22%20d%3D%22M34%201438Q34%201446%2037%201448T50%201450H56H71Q73%201448%2099%201423T144%201380T198%201319T260%201238T323%201137T385%201013T440%20864T485%20688T514%20485T526%20251Q526%20134%20519%2053Q472%20-519%20162%20-860Q139%20-885%20119%20-904T86%20-936T71%20-949H56Q43%20-949%2039%20-947T34%20-937Q88%20-883%20140%20-813Q428%20-430%20428%20251Q428%20453%20402%20628T338%20922T245%201146T145%201309T46%201425Q44%201427%2042%201429T39%201433T36%201436L34%201438Z%22%3E%3C%2Fpath%3E%0A%3Cpath%20stroke-width%3D%221%22%20id%3D%22E1-MJMAIN-64%22%20d%3D%22M376%20495Q376%20511%20376%20535T377%20568Q377%20613%20367%20624T316%20637H298V660Q298%20683%20300%20683L310%20684Q320%20685%20339%20686T376%20688Q393%20689%20413%20690T443%20693T454%20694H457V390Q457%2084%20458%2081Q461%2061%20472%2055T517%2046H535V0Q533%200%20459%20-5T380%20-11H373V44L365%2037Q307%20-11%20235%20-11Q158%20-11%2096%2050T34%20215Q34%20315%2097%20378T244%20442Q319%20442%20376%20393V495ZM373%20342Q328%20405%20260%20405Q211%20405%20173%20369Q146%20341%20139%20305T131%20211Q131%20155%20138%20120T173%2059Q203%2026%20251%2026Q322%2026%20373%20103V342Z%22%3E%3C%2Fpath%3E%0A%3Cpath%20stroke-width%3D%221%22%20id%3D%22E1-MJMAIN-3D%22%20d%3D%22M56%20347Q56%20360%2070%20367H707Q722%20359%20722%20347Q722%20336%20708%20328L390%20327H72Q56%20332%2056%20347ZM56%20153Q56%20168%2072%20173H708Q722%20163%20722%20153Q722%20140%20707%20133H70Q56%20140%2056%20153Z%22%3E%3C%2Fpath%3E%0A%3Cpath%20stroke-width%3D%221%22%20id%3D%22E1-MJMAIN-31%22%20d%3D%22M213%20578L200%20573Q186%20568%20160%20563T102%20556H83V602H102Q149%20604%20189%20617T245%20641T273%20663Q275%20666%20285%20666Q294%20666%20302%20660V361L303%2061Q310%2054%20315%2052T339%2048T401%2046H427V0H416Q395%203%20257%203Q121%203%20100%200H88V46H114Q136%2046%20152%2046T177%2047T193%2050T201%2052T207%2057T213%2061V578Z%22%3E%3C%2Fpath%3E%0A%3Cpath%20stroke-width%3D%221%22%20id%3D%22E1-MJMAIN-2C%22%20d%3D%22M78%2035T78%2060T94%20103T137%20121Q165%20121%20187%2096T210%208Q210%20-27%20201%20-60T180%20-117T154%20-158T130%20-185T117%20-194Q113%20-194%20104%20-185T95%20-172Q95%20-168%20106%20-156T131%20-126T157%20-76T173%20-3V9L172%208Q170%207%20167%206T161%203T152%201T140%200Q113%200%2096%2017Z%22%3E%3C%2Fpath%3E%0A%3Cpath%20stroke-width%3D%221%22%20id%3D%22E1-MJMAIN-2B%22%20d%3D%22M56%20237T56%20250T70%20270H369V420L370%20570Q380%20583%20389%20583Q402%20583%20409%20568V270H707Q722%20262%20722%20250T707%20230H409V-68Q401%20-82%20391%20-82H389H387Q375%20-82%20369%20-68V230H70Q56%20237%2056%20250Z%22%3E%3C%2Fpath%3E%0A%3Cpath%20stroke-width%3D%221%22%20id%3D%22E1-MJMAIN-34%22%20d%3D%22M462%200Q444%203%20333%203Q217%203%20199%200H190V46H221Q241%2046%20248%2046T265%2048T279%2053T286%2061Q287%2063%20287%20115V165H28V211L179%20442Q332%20674%20334%20675Q336%20677%20355%20677H373L379%20671V211H471V165H379V114Q379%2073%20379%2066T385%2054Q393%2047%20442%2046H471V0H462ZM293%20211V545L74%20212L183%20211H293Z%22%3E%3C%2Fpath%3E%0A%3Cpath%20stroke-width%3D%221%22%20id%3D%22E1-MJSZ1-221A%22%20d%3D%22M263%20249Q264%20249%20315%20130T417%20-108T470%20-228L725%20302Q981%20837%20982%20839Q989%20850%201001%20850Q1008%20850%201013%20844T1020%20832V826L741%20243Q645%2043%20540%20-176Q479%20-303%20469%20-324T453%20-348Q449%20-350%20436%20-350L424%20-349L315%20-96Q206%20156%20205%20156L171%20130Q138%20104%20137%20104L111%20130L263%20249Z%22%3E%3C%2Fpath%3E%0A%3Cpath%20stroke-width%3D%221%22%20id%3D%22E1-MJMAIN-2E%22%20d%3D%22M78%2060Q78%2084%2095%20102T138%20120Q162%20120%20180%20104T199%2061Q199%2036%20182%2018T139%200T96%2017T78%2060Z%22%3E%3C%2Fpath%3E%0A%3C%2Fdefs%3E%0A%3Cg%20stroke%3D%22currentColor%22%20fill%3D%22currentColor%22%20stroke-width%3D%220%22%20transform%3D%22matrix(1%200%200%20-1%200%200)%22%20aria-hidden%3D%22true%22%3E%0A%3Cg%20transform%3D%22translate(167%2C0)%22%3E%0A%3Cg%20transform%3D%22translate(-11%2C0)%22%3E%0A%3Cg%20transform%3D%22translate(0%2C1350)%22%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJSZ2-222B%22%20x%3D%220%22%20y%3D%220%22%3E%3C%2Fuse%3E%0A%20%3Cuse%20transform%3D%22scale(0.707)%22%20xlink%3Ahref%3D%22%23E1-MJMAIN-221E%22%20x%3D%221500%22%20y%3D%221540%22%3E%3C%2Fuse%3E%0A%20%3Cuse%20transform%3D%22scale(0.707)%22%20xlink%3Ahref%3D%22%23E1-MJMAIN-30%22%20x%3D%22787%22%20y%3D%22-1270%22%3E%3C%2Fuse%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJMATHI-66%22%20x%3D%222035%22%20y%3D%220%22%3E%3C%2Fuse%3E%0A%3Cg%20transform%3D%22translate(2752%2C0)%22%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJSZ3-28%22%3E%3C%2Fuse%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJMAIN-28%22%20x%3D%22736%22%20y%3D%220%22%3E%3C%2Fuse%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJMATHI-61%22%20x%3D%221126%22%20y%3D%220%22%3E%3C%2Fuse%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJMATHI-78%22%20x%3D%221655%22%20y%3D%220%22%3E%3C%2Fuse%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJMAIN-2212%22%20x%3D%222450%22%20y%3D%220%22%3E%3C%2Fuse%3E%0A%3Cg%20transform%3D%22translate(3228%2C0)%22%3E%0A%3Cg%20transform%3D%22translate(342%2C0)%22%3E%0A%3Crect%20stroke%3D%22none%22%20width%3D%22692%22%20height%3D%2260%22%20x%3D%220%22%20y%3D%22220%22%3E%3C%2Frect%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJMATHI-62%22%20x%3D%22131%22%20y%3D%22676%22%3E%3C%2Fuse%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJMATHI-78%22%20x%3D%2260%22%20y%3D%22-686%22%3E%3C%2Fuse%3E%0A%3C%2Fg%3E%0A%3C%2Fg%3E%0A%3Cg%20transform%3D%22translate(4383%2C0)%22%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJMAIN-29%22%20x%3D%220%22%20y%3D%220%22%3E%3C%2Fuse%3E%0A%20%3Cuse%20transform%3D%22scale(0.707)%22%20xlink%3Ahref%3D%22%23E1-MJMAIN-32%22%20x%3D%22550%22%20y%3D%22583%22%3E%3C%2Fuse%3E%0A%3C%2Fg%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJSZ3-29%22%20x%3D%225226%22%20y%3D%22-1%22%3E%3C%2Fuse%3E%0A%3C%2Fg%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJMAIN-64%22%20x%3D%228882%22%20y%3D%220%22%3E%3C%2Fuse%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJMATHI-78%22%20x%3D%229438%22%20y%3D%220%22%3E%3C%2Fuse%3E%0A%3C%2Fg%3E%0A%3Cg%20transform%3D%22translate(1232%2C-1351)%22%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJSZ2-222B%22%20x%3D%220%22%20y%3D%220%22%3E%3C%2Fuse%3E%0A%20%3Cuse%20transform%3D%22scale(0.707)%22%20xlink%3Ahref%3D%22%23E1-MJMAIN-221E%22%20x%3D%221500%22%20y%3D%221540%22%3E%3C%2Fuse%3E%0A%20%3Cuse%20transform%3D%22scale(0.707)%22%20xlink%3Ahref%3D%22%23E1-MJMAIN-30%22%20x%3D%22787%22%20y%3D%22-1270%22%3E%3C%2Fuse%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJMATHI-66%22%20x%3D%222035%22%20y%3D%220%22%3E%3C%2Fuse%3E%0A%3Cg%20transform%3D%22translate(2752%2C0)%22%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJSZ3-28%22%3E%3C%2Fuse%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJMATHI-61%22%20x%3D%22736%22%20y%3D%220%22%3E%3C%2Fuse%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJMATHI-78%22%20x%3D%221266%22%20y%3D%220%22%3E%3C%2Fuse%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJMAIN-2B%22%20x%3D%222060%22%20y%3D%220%22%3E%3C%2Fuse%3E%0A%3Cg%20transform%3D%22translate(2839%2C0)%22%3E%0A%3Cg%20transform%3D%22translate(342%2C0)%22%3E%0A%3Crect%20stroke%3D%22none%22%20width%3D%22692%22%20height%3D%2260%22%20x%3D%220%22%20y%3D%22220%22%3E%3C%2Frect%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJMATHI-62%22%20x%3D%22131%22%20y%3D%22676%22%3E%3C%2Fuse%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJMATHI-78%22%20x%3D%2260%22%20y%3D%22-686%22%3E%3C%2Fuse%3E%0A%3C%2Fg%3E%0A%3C%2Fg%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJSZ3-29%22%20x%3D%223993%22%20y%3D%22-1%22%3E%3C%2Fuse%3E%0A%3C%2Fg%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJMAIN-64%22%20x%3D%227649%22%20y%3D%220%22%3E%3C%2Fuse%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJMATHI-78%22%20x%3D%228205%22%20y%3D%220%22%3E%3C%2Fuse%3E%0A%3C%2Fg%3E%0A%3C%2Fg%3E%0A%3Cg%20transform%3D%22translate(10000%2C0)%22%3E%0A%3Cg%20transform%3D%22translate(0%2C1350)%22%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJMAIN-3D%22%20x%3D%22277%22%20y%3D%220%22%3E%3C%2Fuse%3E%0A%3Cg%20transform%3D%22translate(1056%2C0)%22%3E%0A%3Cg%20transform%3D%22translate(397%2C0)%22%3E%0A%3Crect%20stroke%3D%22none%22%20width%3D%22649%22%20height%3D%2260%22%20x%3D%220%22%20y%3D%22220%22%3E%3C%2Frect%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJMAIN-31%22%20x%3D%2274%22%20y%3D%22676%22%3E%3C%2Fuse%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJMATHI-61%22%20x%3D%2260%22%20y%3D%22-686%22%3E%3C%2Fuse%3E%0A%3C%2Fg%3E%0A%3C%2Fg%3E%0A%3Cg%20transform%3D%22translate(2390%2C0)%22%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJSZ2-222B%22%20x%3D%220%22%20y%3D%220%22%3E%3C%2Fuse%3E%0A%20%3Cuse%20transform%3D%22scale(0.707)%22%20xlink%3Ahref%3D%22%23E1-MJMAIN-221E%22%20x%3D%221500%22%20y%3D%221540%22%3E%3C%2Fuse%3E%0A%20%3Cuse%20transform%3D%22scale(0.707)%22%20xlink%3Ahref%3D%22%23E1-MJMAIN-30%22%20x%3D%22787%22%20y%3D%22-1270%22%3E%3C%2Fuse%3E%0A%3C%2Fg%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJMATHI-66%22%20x%3D%224425%22%20y%3D%220%22%3E%3C%2Fuse%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJMAIN-28%22%20x%3D%224975%22%20y%3D%220%22%3E%3C%2Fuse%3E%0A%3Cg%20transform%3D%22translate(5365%2C0)%22%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJMATHI-78%22%20x%3D%220%22%20y%3D%220%22%3E%3C%2Fuse%3E%0A%20%3Cuse%20transform%3D%22scale(0.707)%22%20xlink%3Ahref%3D%22%23E1-MJMAIN-32%22%20x%3D%22809%22%20y%3D%22583%22%3E%3C%2Fuse%3E%0A%3C%2Fg%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJMAIN-29%22%20x%3D%226391%22%20y%3D%220%22%3E%3C%2Fuse%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJMAIN-64%22%20x%3D%226781%22%20y%3D%220%22%3E%3C%2Fuse%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJMATHI-78%22%20x%3D%227337%22%20y%3D%220%22%3E%3C%2Fuse%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJMAIN-2C%22%20x%3D%227910%22%20y%3D%220%22%3E%3C%2Fuse%3E%0A%3C%2Fg%3E%0A%3Cg%20transform%3D%22translate(0%2C-1351)%22%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJMAIN-3D%22%20x%3D%22277%22%20y%3D%220%22%3E%3C%2Fuse%3E%0A%3Cg%20transform%3D%22translate(1056%2C0)%22%3E%0A%3Cg%20transform%3D%22translate(397%2C0)%22%3E%0A%3Crect%20stroke%3D%22none%22%20width%3D%22649%22%20height%3D%2260%22%20x%3D%220%22%20y%3D%22220%22%3E%3C%2Frect%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJMAIN-31%22%20x%3D%2274%22%20y%3D%22676%22%3E%3C%2Fuse%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJMATHI-61%22%20x%3D%2260%22%20y%3D%22-686%22%3E%3C%2Fuse%3E%0A%3C%2Fg%3E%0A%3C%2Fg%3E%0A%3Cg%20transform%3D%22translate(2390%2C0)%22%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJSZ2-222B%22%20x%3D%220%22%20y%3D%220%22%3E%3C%2Fuse%3E%0A%20%3Cuse%20transform%3D%22scale(0.707)%22%20xlink%3Ahref%3D%22%23E1-MJMAIN-221E%22%20x%3D%221500%22%20y%3D%221540%22%3E%3C%2Fuse%3E%0A%20%3Cuse%20transform%3D%22scale(0.707)%22%20xlink%3Ahref%3D%22%23E1-MJMAIN-30%22%20x%3D%22787%22%20y%3D%22-1270%22%3E%3C%2Fuse%3E%0A%3C%2Fg%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJMATHI-66%22%20x%3D%224425%22%20y%3D%220%22%3E%3C%2Fuse%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJMAIN-28%22%20x%3D%224975%22%20y%3D%220%22%3E%3C%2Fuse%3E%0A%3Cg%20transform%3D%22translate(5365%2C0)%22%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJSZ1-221A%22%20x%3D%220%22%20y%3D%22126%22%3E%3C%2Fuse%3E%0A%3Crect%20stroke%3D%22none%22%20width%3D%223708%22%20height%3D%2260%22%20x%3D%221000%22%20y%3D%22917%22%3E%3C%2Frect%3E%0A%3Cg%20transform%3D%22translate(1000%2C0)%22%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJMATHI-78%22%20x%3D%220%22%20y%3D%220%22%3E%3C%2Fuse%3E%0A%20%3Cuse%20transform%3D%22scale(0.707)%22%20xlink%3Ahref%3D%22%23E1-MJMAIN-32%22%20x%3D%22809%22%20y%3D%22583%22%3E%3C%2Fuse%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJMAIN-2B%22%20x%3D%221248%22%20y%3D%220%22%3E%3C%2Fuse%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJMAIN-34%22%20x%3D%222249%22%20y%3D%220%22%3E%3C%2Fuse%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJMATHI-61%22%20x%3D%222749%22%20y%3D%220%22%3E%3C%2Fuse%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJMATHI-62%22%20x%3D%223279%22%20y%3D%220%22%3E%3C%2Fuse%3E%0A%3C%2Fg%3E%0A%3C%2Fg%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJMAIN-29%22%20x%3D%2210074%22%20y%3D%220%22%3E%3C%2Fuse%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJMAIN-64%22%20x%3D%2210464%22%20y%3D%220%22%3E%3C%2Fuse%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJMATHI-78%22%20x%3D%2211020%22%20y%3D%220%22%3E%3C%2Fuse%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJMAIN-2E%22%20x%3D%2211593%22%20y%3D%220%22%3E%3C%2Fuse%3E%0A%3C%2Fg%3E%0A%3C%2Fg%3E%0A%3C%2Fg%3E%0A%3C%2Fg%3E%0A%3C%2Fsvg%3E#card=math&code=%5Cbegin%7Balign%7D%0A%5Cint_0%5E%5Cinfty%20f%5Cleft%28%20%28ax-%5Cfrac%7Bb%7D%7Bx%7D%29%5E2%20%5Cright%29%5Cmathrm%7Bd%7D%20x%20%26%3D%20%5Cfrac%7B1%7D%7Ba%7D%5Cint_0%5E%5Cinfty%20f%28x%5E2%29%5Cmathrm%7Bd%7D%20x%2C%20%5C%5C%0A%5Cint_0%5E%5Cinfty%20f%5Cleft%28ax%2B%20%5Cfrac%7Bb%7D%7Bx%7D%5Cright%29%20%5Cmathrm%7Bd%7D%20x%20%26%3D%20%5Cfrac%7B1%7D%7Ba%7D%20%5Cint_0%5E%7B%5Cinfty%7Df%28%5Csqrt%7Bx%5E2%2B4ab%7D%29%20%5Cmathrm%7Bd%7D%20x.%0A%5Cend%7Balign%7D%0A&id=LdvlG)

（我们假设上述反常积分都是收敛的.）

证明：(1)先证明第一个等式. 设. 左边=%5E2%20%5Cright)%20%5Cmathrm%7Bd%7Dy%20%3D%20I_1%20%2B%20I_2#card=math&code=%5Csqrt%7B%5Cfrac%7Bb%7D%7Ba%7D%7D%20%5Cint_0%5E%5Cinfty%20f%5Cleft%28ab%20%28y-%5Cfrac%7B1%7D%7By%7D%29%5E2%20%5Cright%29%20%5Cmathrm%7Bd%7Dy%20%3D%20I_1%20%2B%20I_2&id=EPRck), 其中分别是被积函数在与#card=math&code=%5B1%2C%20%5Cinfty%29&id=BIQix)上的积分.

%5E2%20%5Cright)%20%5Cmathrm%7Bd%7Dy%20%3D%20%5Csqrt%7B%5Cfrac%7Bb%7D%7Ba%7D%7D%20%5Cint_1%5E%5Cinfty%20f%5Cleft(%20ab%20(t%20-%20%5Cfrac%7B1%7D%7Bt%7D)%5E2%5Cright)%20%5Cfrac%7B%5Cmathrm%7Bd%7Dt%7D%7Bt%5E2%7D#card=math&code=I_1%20%3D%20%5Csqrt%7B%5Cfrac%7Bb%7D%7Ba%7D%7D%5Cint_0%5E1%20f%5Cleft%28ab%20%28y-%5Cfrac%7B1%7D%7By%7D%29%5E2%20%5Cright%29%20%5Cmathrm%7Bd%7Dy%20%3D%20%5Csqrt%7B%5Cfrac%7Bb%7D%7Ba%7D%7D%20%5Cint_1%5E%5Cinfty%20f%5Cleft%28%20ab%20%28t%20-%20%5Cfrac%7B1%7D%7Bt%7D%29%5E2%5Cright%29%20%5Cfrac%7B%5Cmathrm%7Bd%7Dt%7D%7Bt%5E2%7D&id=oulP3).
其中第二个等号是通过变换而得到——这是处理反常积分的常用技巧.

于是原积分等于%5E2%5Cright)(%5Cmathrm%7Bd%7Dt%2B%5Cfrac%7B%5Cmathrm%7Bd%7Dt%7D%7Bt%5E2%7D)#card=math&code=I_1%2BI_2%20%3D%20%5Csqrt%7B%5Cfrac%7Bb%7D%7Ba%7D%7D%20%5Cint_1%5E%5Cinfty%20f%5Cleft%28ab%20%28t-%5Cfrac%7B1%7D%7Bt%7D%29%5E2%5Cright%29%28%5Cmathrm%7Bd%7Dt%2B%5Cfrac%7B%5Cmathrm%7Bd%7Dt%7D%7Bt%5E2%7D%29&id=BsUNk) , 注意#card=math&code=%5Cmathrm%7Bd%7Dt%2B%20%5Cfrac%7B%5Cmathrm%7Bd%7Dt%7D%7Bt%5E2%7D%20%3D%20%5Cmathrm%7Bd%7D%28t-%5Cfrac%7B1%7D%7Bt%7D%29&id=Ido9h). 所以

%5E2%20)%5Cmathrm%7Bd%7D(t-%20%5Cfrac%7B1%7D%7Bt%7D)%20%3D%20%5Csqrt%7B%5Cfrac%7Bb%7D%7Ba%7D%7D%20%5Cint_0%5E%5Cinfty%20f(abz%5E2)%5Cmathrm%7Bd%7Dz#card=math&code=I_1%20%2B%20I_2%20%3D%20%5Csqrt%7B%5Cfrac%7Bb%7D%7Ba%7D%7D%5Cint_1%5E%5Cinfty%20f%28ab%20%28t-%5Cfrac%7B1%7D%7Bt%7D%29%5E2%20%29%5Cmathrm%7Bd%7D%28t-%20%5Cfrac%7B1%7D%7Bt%7D%29%20%3D%20%5Csqrt%7B%5Cfrac%7Bb%7D%7Ba%7D%7D%20%5Cint_0%5E%5Cinfty%20f%28abz%5E2%29%5Cmathrm%7Bd%7Dz&id=TPPGv). 最后一步是用代换, 而在时单调递增.

最后设, 得到.

(2)可以类似(1)来证明. 为简便，可以先考虑的情形，再作代换.

应用

结合Gauss积分 , 可以用上面公式计算下面两个积分.

1. 
2. %7D%20%5Csqrt%7Bx%7D%7D#card=math&code=%5Cdisplaystyle%20%5Cint_%7B0%7D%5E%7B%5Cinfty%7D%20%5Cfrac%7B%5Cmathrm%7Bd%7Dx%7D%7Be%5E%7B1985%28x%2B%5Cfrac%7B1%7D%7Bx%7D%29%7D%20%5Csqrt%7Bx%7D%7D&id=eW8Nk). 作代换, 原积分可化为

最后，我们指出，表达式与双曲余弦、双曲正弦有关. 请看后续笔记.