积分表127个公式的推导(修正版)

高等数学积分表公式推导目录（一）含有bax的积分（1~9）·······················································1（二）含有bax的积分（10~18）···················································5（三）含有22ax的积分（19~21）····················································9（四）含有)0(2abax的积分（22~28）············································11（五）含有)0(2acbxax的积分（29~30）········································14（六）含有)0(22aax的积分（31~44）·········································15（七）含有)0(22aax的积分（45~58）·········································24（八）含有)0(22axa的积分（59~72）·········································37（九）含有)0(2acbxa的积分（73~78）····································48（十）含有或))((xbax的积分（79~82）···························51（十一）含有三角函数的积分（83~112）···········································55（十二）含有反三角函数的积分（其中0a)（113~121）·······················68（十三）含有指数函数的积分（122~131）··········································73（十四）含有对数函数的积分（132~136）··········································78（十五）含有双曲函数的积分（137~141）··········································80（十六）定积分（142~147）····························································81附录：常数和基本初等函数导数公式·········································85说明·····················································································86bxax-1-（一）含有bax的积分（1~9）CbaxlnabaxdxbaxtCtlnadttabaxdxdtadx,adxdtttbaxabxxbax)x(fCbaxlnabaxdx.11111)0(}|{111代入上式得：将，则令的定义域为被积函数证明：CbaxμadxbaxbaxtCtμadttadxbaxdtadx,adxdttbaxμCbaxμadxbax.μμμμμμμ111)()1(1)()1(11)(1,1)()()1(1)(2代入上式得：将则令证明：CbaxlnbbaxadxbaxxbaxtCtlnbtaCtlnabatdttbadtadttb1adta·tbtadxbaxxdtadx,btax,ttbaxabx|xbaxx)x(fCbaxlnbbaxadxbaxx.22222222111111111)0(}{13代入上式得：将则令的定义域为被积函数证明：-2-CbaxlnbbaxbbaxadxbaxxCbaxlnabbaxdbaxabdxbaxbaCbaxlnabxabbaxdbaxabdxabaxdbaxbbaxabdxbaxabxaCbaxadxbaxadxbaxbadxbaxabxadxbaxadxbaxbabxbaxadxbaxxCbaxlnbbaxbbaxadxbaxx)(2)(211)(1122)(122)(221)(21)(1121)(1)2)(1)(2)(211.4223233232222323323321232222222222232由以上各式整理得：证明：CxbaxlnbCbaxxlnbCbaxlnbxlnb)bax(dbaxbdxxbdxbaxbadxxbdx)bax(babxbaxxdxbabAbBAabxaxbaxbaxBxbaxxabx|xbaxx)x(fCxbaxlnbbaxxdx.11111111111]1[)(B1A10AB)(AB)A(1,A)(1}{)(11)(5于是有则设的定义域为被积函数证明：blogblogaa1提示：-3-CxbaxlnbabxCbaxlnbabxxlnbabaxdbaxbadxxbdxxbadxbaxbadxxbdxxbabaxxdxbaCbbaBbaBAbCAabaBAbxaxCxbaxbaxxbaxCxBxbaxxabxxbaxxxfCxbaxlnbabxbaxxdx11)(11111111)(1BA1001B)(C)(A)B()(A1,A)(1}|{)(1)(1)(.6222222222222222222222于是有即则设的定义域为被积函数证明：CbaxbbaxlnaCbaxabbaxlnabaxdbaxabbaxdbaxadxbaxabdxbaxadxbaxxabBaBAbAaxBAbaxbaxxbaxBbaxAbaxxabx|xbaxx)x(fCbaxbbaxlnadxbaxx.1)(1)()(1)(11)(111)(1A01)(AB)A(,)()(}{)(1)(72222222222222于是有即则设的定义域为被积函数证明：-4-CbaxbbaxlnbbaxadxbaxxbaxtCtbtlnbtaCtlnabtatabdttabdtadttabdttabttbdxbaxxtabttbtatbbaxxdtadx,btax,ttbaxabx|xbaxx)x(fCbaxbbaxlnbbaxadxbaxx.23222333323323223222222222222222232221)()2(12112112)(2)()(11)0(}{)(21)(8代入上式得：将则令的定义域为被积函数证明：C|xbax|ln·bbaxbCbax·bb||axlnb|x|lnbdxbaxbadxbaxbadxxbbaxxdxbaDbaBbA1Ab0DBbAab20BaAaAbDBbAab2xBaAaxDxBbxBaxAabx2AbxAaDxbaxBxbaxA1baxDbaxBxAbaxxabx|xbaxx)x(fC|xbax|lnbbaxbbaxxdx.222222222222222222222221)(11111)(1111)(1)()()()()()(1}{)(1·1)(1)(9于是有则设：的定义域为证明：被积函数-5-（二）含有bax的积分（10~18）CbaxaCbaxabaxdbaxadxbaxCbaxadxbax3121213)(32)(21111)()(1)(32.10证明：CbaxbaxaCbaxbbaxadxbaxxbaxtCbtatCtabtadtabdtadtbttadtattabtdxbaxxtabtbaxxdtatdxabtxttbaxCbaxbaxadxbaxx32322233252325224222232)()23(152)(]5)(3[152)53(15232523252)(22,2,,)0()()23(152.11代入上式得：将则令证明：CbaxbabxxaabaxbbabxbxabaxadxbaxxbaxtCbtbtatCtabtabtaCtabtabtadttabdttabdttadtbttbttadxbaxxabttbttabtbaxxdtatdxabtxttbaxCbaxbabxxaadxbaxx3222322223322243353332731432132163432326332532232522222322232)()81215(1052)(4235301515)(1052)423515(1052543272411421126112422)2(22)(,2,,)0()()81215(1052.12代入上式得：将则令证明：-6-CbaxbaxaCbaxabbaxbaxadxbaxxbaxtCtabtaCtabtabdtadttadtatatbtdxbaxxdtatdxabtxttbaxCbaxbaxadxbaxx)()2(32)(2)()(3223222112222,2,,)0()()2(32.132222322122222222代入上式得：将则令证明：CbaxbabxxaaCbaxbaxbbabxbxabaxadxbaxxbaxtCbtbtatCtbtbtadttabdtbadttadtbtbtadtattabtdxbaxxdtatdxabtxttbaxCbaxbabxxaadxbaxx)()843(152)()(1015)2(3)(152)10153(152)3251(2422)2(221)(,2,,)0()()843(152.142223222232224332532323432243222222232代入上式得：将则令证明：-7-