高等数学讲义之积分表公式推导

高等数学高等数学高等数学高等数学积积积积分分分分表表表表公公公公式式式式推推推推导导导导目目目目录录录录（一）含有（一）含有（一）含有（一）含有bax+的积分的积分的积分的积分（1~9）·······················································1（二）含有（二）含有（二）含有（二）含有bax+的积分的积分的积分的积分（10~18）···················································5（三）含有（三）含有（三）含有（三）含有22ax±的积分的积分的积分的积分（19~21）····················································9（四）含有（四）含有（四）含有（四）含有)0(2+abax的积分的积分的积分的积分（22~28）············································11（五）含有（五）含有（五）含有（五）含有)0(2++acbxax的积分的积分的积分的积分（29~30）········································14（六）含有（六）含有（六）含有（六）含有)0(22+aax的积分的积分的积分的积分（31~44）·········································15（七）含有（七）含有（七）含有（七）含有)0(22−aax的积分的积分的积分的积分（45~58）·········································24（八）含有（八）含有（八）含有（八）含有)0(22−axa的积分的积分的积分的积分（59~72）·········································37（九）含有（九）含有（九）含有（九）含有)0(2++±acbxa的积分的积分的积分的积分（73~78）····································48（十）含有（十）含有（十）含有（十）含有或或或或))((xbax−−的积分的积分的积分的积分（79~82）···························51（十一）含有三角函数的积分（十一）含有三角函数的积分（十一）含有三角函数的积分（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（十二）含有反三角函数的积分（其中（十二）含有反三角函数的积分（其中（十二）含有反三角函数的积分（其中0a))))（113~121）·······················68（十三）含有指数函数的积分（十三）含有指数函数的积分（十三）含有指数函数的积分（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（十四）含有对数函数的积分（十四）含有对数函数的积分（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（十五）含有双曲函数的积分（十五）含有双曲函数的积分（十五）含有双曲函数的积分（137~141）··········································80（十六）定积分（十六）定积分（十六）定积分（十六）定积分（142~147）····························································81附录：常数和基本初等函数导数公式附录：常数和基本初等函数导数公式附录：常数和基本初等函数导数公式附录：常数和基本初等函数导数公式·········································85说明说明说明说明·····················································································86团队人员团队人员团队人员团队人员··············································································87bxax−−±-1-（一）含有（一）含有（一）含有（一）含有bax+的积分的积分的积分的积分（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（二）含有（二）含有（二）含有bax+的积分的积分的积分的积分（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