2022年《高等数学同济五版》讲稿WORD版-第04章不定积分 .pdf

第四章不定积分教学目的：1、理解原函数概念、不定积分的概念。2、掌握不定积分的基本公式，掌握不定积分的性质，掌握换元积分法（第一，第二）与分部积分法。3、会求有理函数、三角函数有理式和简单无理函数的积分。教学重点：1、不定积分的概念；2、不定积分的性质及基本公式；3、换元积分法与分部积分法。教学难点：1、换元积分法；2、分部积分法；3、三角函数有理式的积分。4 1 不定积分的概念与性质一、原函数与不定积分的概念定义 1如果在区间I 上可导函数F(x)的导函数为f(x)即对任一x I都有F(x)f(x)或 dF(x)f(x)dx那么函数F(x)就称为 f(x)(或 f(x)dx)在区间 I 上的原函数例如因为(sin x)cos x所以 sin x 是 cos x 的原函数又如当 x(1)时因为xx21)(所以x 是x21的原函数提问:cos x 和x21还有其它原函数吗？原函数存在定理如果函数 f(x)在区间 I 上连续那么在区间I 上存在可导函数F(x)使对任一 xI 都有F(x)f(x)简单地说就是连续函数一定有原函数两点说明第一如果函数 f(x)在区间 I 上有原函数F(x)那么 f(x)就有无限多个原函数F(x)C 都是 f(x)的原函数其中 C 是任意常数第二f(x)的任意两个原函数之间只差一个常数即如果(x)和 F(x)都是 f(x)的原函数则(x)F(x)C(C 为某个常数)定义 2 在区间 I 上函数 f(x)的带有任意常数项的原函数称为f(x)(或 f(x)dx)在区间 I 上的不定积分记作dxxf)(其中记号称为积分号f(x)称为被积函数f(x)dx 称为被积表达式x 称为积分变量根据定义如果 F(x)是 f(x)在区间 I 上的一个原函数那么 F(x)C 就是 f(x)的不定积分即CxFdxxf)()(因而不定积分dxxf)(可以表示f(x)的任意一个原函数例 1 因为 sin x 是 cos x 的原函数所以Cxxdxsincos因为x 是x21的原函数所以Cxdxx21例 2.求函数xxf1)(的不定积分解：当 x0 时(ln x)x1Cxdxxln1(x0)当 x0 时ln(x)xx1)1(1Cxdxx)ln(1(x0)解:设 x a sin t22t那么22xatataacossin222dxa cos t d t于是tdtatadxxacoscos22Cttatdta)2sin4121(cos222因为axtarcsin,axaaxttt222cossin22sin所以dxxa22Ctta)2sin4121(2Cxaxaxa22221arcsin2解:设 x a sin t22t那么tdtatadxxacoscos22Cttatdta)2sin4121(cos222Cxaxaxa22221arcsin2提示:22xatataacossin222dx acos tdt提示:axtarcsin,axaaxttt222cossin22sin例 20.求22axdx(a0)解法一设 x a tan t22t那么22axtaa222tanta2tan1a sec tdx a sec 2t d t于是文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z822axdxtdtdttatasecsecsec2 ln|sec t tan t|C因为aaxt22secaxttan所以22axdx ln|sec t tan t|CCaaxax)ln(22122)ln(Caxx其中 C 1C ln a解法一设 x a tan t22t那么tdtdttataaxdxsecsecsec222ln|sect tant|CCaaxax)ln(22122)ln(Caxx其中 C 1C ln a提示:22axtaa222tanasectdx a sec 2t dt提示:aaxt22secaxttan解法二:设 x a sh t那么22axdxCaxCtdtdttataarshch ch Caxax1)(ln2122)ln(Caxx其中 C 1C ln a提示:22ax222atshaa ch tdxa ch t d t例 23.求22axdx(a0)解:当 xa 时设 x a sec t(20 t)那么22ax222secata1sec2taa tan t于是文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z822axdxtdtdttattasectantansec ln|sec t tan t|C因为aaxt22tanaxtsec所以22axdx ln|sec t tan t|CCaaxax|ln22122)ln(Caxx其中 C 1C ln a当 xa 于是22axdxCauuaudu)ln(2222Caxx)ln(22122)ln(Caxx122222)ln(lnCaxxCaaxx其中 C 1C 2ln a综合起来有22axdxCaxx|ln22解:当 xa 时设 x a sec t(20 t)那么22axdxt d tdttattasectantansecCaaxaxCtt)l n(|t ansec|ln22Caxx)l n(22其中 C 1C ln a当 xa于是22axdxCauuaudu)ln(2222CaaxxCaxx22222ln)ln(122)ln(Caxx其中 C 1C 2ln a提示:22ax222secata1sec2taatant文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8提示:aaxt22tanaxtsec综合起来有Caxxaxdx|ln2222补充公式(16)Cxxdx|cos|lntanCxxdx|sin|lncot(18)Cxxxdx|tansec|lnsec(19)Cxxxdx|cotcsc|lncsc(20)Caxadxxaarctan1122(21)Caxaxadxax|ln21122(22)Caxdxxaarcsin122(23)Caxxaxdx)ln(2222(24)Caxxaxdx|ln2222 4 3 分部积分法设函数 u u(x)及 v v(x)具有连续导数那么两个函数乘积的导数公式为(uv)u v uv移项得uv(uv)u v对这个等式两边求不定积分得vdxuuvdxvu或vduuvudv这个公式称为分部积分公式分部积分过程:vdxuuvvduuvudvdxvu例 1 xdxxxxxdxdxxsinsinsincosx sin x cos x C文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8例 2 Cexedxexexdedxxexxxxxx例 3 2222dxeexdexdxexxxxxxxxxxdeexdxxeex2222dxexeexxxx222x2ex2xex2exCex(x22x 2)C例 4 dxxxxxxdxxdxx121ln21ln21ln222Cxxxxdxxx22241ln2121ln21例 5 xxdxxxdxarccosarccosarccosdxxxxx211arccos)1()1(21arccos2212xdxxxCxxx21arccos例 6 2arctan21arctanxdxxdxxdxxxxx2221121arctan21dxxxx)111(21arctan2122Cxxxxarctan2121arctan212例 7 求xdxexsin解 因为xdexexdexdxexxxxsinsinsinsinxxxxxdexexdxexecossincossinxdexexexxxcoscossinxdexexexxxcoscossinxdxexexexxxsincossin所以Cxxexdxexx)cos(sin21sin例 8 求xdx3sec文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8解 因为xxdxdxxxdxtansecsecsecsec23xdxxxx2tansectansecdxxxxx)1(secsectansec2xdxxdxxxsecsectansec3xdxxxxx3sec|tansec|lntansec所以xdx3secCxxxx|)tansec|lntan(sec21例 9 求nnaxdxI)(22其中 n 为正整数解CaxaaxdxIarctan1221当 n 1 时，用分部积分法有dxaxxnaxxaxdxnnn)()1(2)()(222122122dxaxaaxnaxxnnn)()(1)1(2)(222122122即)(1(2)(211221nnnnIaInaxxI于是)32()()1(2111222nnnInaxxnaI以此作为递推公式并由CaxaIarctan11即可得nI例 10 求dxex解 令 xt 2则dx 2tdt于dxexCxeCtedttextt)1(2)1(22xdexxdedxexxx2)(2xdeexdexxxx222CxeCeexxxx)1(222第一换元法与分部积分法的比较:文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P4J1Z8文档编码:CU6W5A3V4U6 HL3L2K8A3G5 ZI3P1P