常用积分公式汇总+较全面的积分表[1]

INTEGRALSELEMENTARYFORMS1.Zadx¼ax2.ZafðxÞdx¼aZfðxÞdx3.ZðyÞdx¼ZðyÞy0dy,wherey0¼dydx4.ZðuþvÞdx¼ZudxþZvdx,whereuandvareanyfunctionsofx5.Zudv¼uZdvZvdu¼uvZvdu6.Zudvdxdx¼uvZvdudxdx7.Zxndx¼xnþ1nþ1,exceptn¼18.Zf0ðxÞdxfðxÞ¼logfðxÞ,ðdfðxÞ¼f0ðxÞdxÞ9.Zdxx¼logx10.Zf0ðxÞdx2ffiffiffiffiffiffiffiffiffifðxÞp¼ffiffiffiffiffiffiffiffiffifðxÞp,ðdfðxÞ¼f0ðxÞdxÞ11.Zexdx¼ex12.Zeaxdx¼eax=a13.Zbaxdx¼baxalogb,ðb0Þ14.Zlogxdx¼xlogxx15.Zaxlogadx¼ax,ða0Þ16.Zdxa2þx2¼1atan1xa17.Zdxa2x2¼1atanh1xaor12alogaþxax,ða2x2Þ8:18.Zdxx2a2¼1acoth1xaor12alogxaxþa,ðx2a2Þ8:A-20INTEGRALS(Continued)19.Zdxffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia2x2p¼sin1xjajorcos1xjaj,ða2x2Þ8:20.Zdxffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix2a2p¼logðxþffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix2a2pÞ21.Zdxxffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix2a2p¼1jajsec1xa22.Zdxxffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia2x2p¼1alogaþffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia2x2px!FORMSCONTAINING(aþbx)Forformscontainingaþbx,butnotlistedinthetable,thesubstitutionu¼aþbxxmayprovehelpful.23.ZðaþbxÞndx¼ðaþbxÞnþ1ðnþ1Þb,ðn6¼1Þ24.ZxðaþbxÞndx¼1b2ðnþ2ÞðaþbxÞnþ2ab2ðnþ1ÞðaþbxÞnþ1,ðn6¼1,2Þ25.Zx2ðaþbxÞndx¼1b3ðaþbxÞnþ3nþ32aðaþbxÞnþ2nþ2þa2ðaþbxÞnþ1nþ1 26.ZxmðaþbxÞndx¼xmþ1ðaþbxÞnmþnþ1þanmþnþ1ZxmðaþbxÞn1dxor1aðnþ1Þxmþ1ðaþbxÞnþ1þðmþnþ2ÞZxmðaþbxÞnþ1dx or1bðmþnþ1ÞxmðaþbxÞnþ1maZxm1ðaþbxÞndx 8:27.Zdxaþbx¼1blogðaþbxÞ28.ZdxðaþbxÞ2¼1bðaþbxÞ29.ZdxðaþbxÞ3¼12bðaþbxÞ230.Zxdxaþbx¼1b2½aþbxalogðaþbxÞorxbab2logðaþbxÞ8:A-21INTEGRALS(Continued)31.ZxdxðaþbxÞ2¼1b2logðaþbxÞþaaþbx 32.ZxdxðaþbxÞn¼1b21ðn2ÞðaþbxÞn2þaðn1ÞðaþbxÞn1 ,n6¼1,233.Zx2dxaþbx¼1b312ðaþbxÞ22aðaþbxÞþa2logðaþbxÞ 34.Zx2dxðaþbxÞ2¼1b3aþbx2alogðaþbxÞa2aþbx 35.Zx2dxðaþbxÞ3¼1b3logðaþbxÞþ2aaþbxa22ðaþbxÞ2 36.Zx2dxðaþbxÞn¼1b31ðn3ÞðaþbxÞn3þ2aðn2ÞðaþbxÞn2a2ðn1ÞðaþbxÞn1 ,n6¼1,2,337.ZdxxðaþbxÞ¼1alogaþbxx38.ZdxxðaþbxÞ2¼1aðaþbxÞ1a2logaþbxx39.ZdxxðaþbxÞ3¼1a3122aþbxaþbx2þlogxaþbx#40.Zdxx2ðaþbxÞ¼1axþba2logaþbxx41.Zdxx3ðaþbxÞ¼2bxa2a2x2þb2a3logxaþbx42.Zdxx2ðaþbxÞ2¼aþ2bxa2xðaþbxÞþ2ba3logaþbxxFORMSCONTAININGc2x2,x2c243.Zdxc2þx2¼1ctan1xc44.Zdxc2x2¼12clogcþxcx,ðc2x2Þ45.Zdxx2c2¼12clogxcxþc,ðx2c2Þ46.Zxdxc2x2¼12logðc2x2Þ47.Zxdxðc2x2Þnþ1¼12nðc2x2Þn48.Zdxðc2x2Þn¼12c2ðn1Þxðc2x2Þn1þð2n3ÞZdxðc2x2Þn1 49.Zdxðx2c2Þn¼12c2ðn1Þxðx2c2Þn1ð2n3ÞZdxðx2c2Þn1 50.Zxdxx2c2¼12logðx2c2ÞA-22INTEGRALS(Continued)51.Zxdxðx2c2Þnþ1¼12nðx2c2ÞnFORMSCONTAININGaþbxandcþdxu¼aþbx,v¼cþdx,k¼adbcIfk¼0,thenv¼cau52.Zdxuv¼1klogvu53.Zxdxuv¼1kablogðuÞcdlogðvÞhi54.Zdxu2v¼1k1uþdklogvu55.Zxdxu2v¼abkuck2logvu56.Zx2dxu2v¼a2b2kuþ1k2c2dlogðvÞþaðkbcÞb2logðuÞ 57.Zdxunvm¼1kðm1Þ1un1vm1ðmþn2ÞbZdxunvm1 58.Zuvdx¼bxdþkd2logðvÞ59.Zumdxvn¼1kðn1Þumþ1vn1þbðnm2ÞZumvn1dx or1dðnm1Þumvn1þmkZum1vndx or1dðn1Þumvn1mbZum1vn1dx 8:FORMSCONTAINING(aþbxn)60.Zdxaþbx2¼1ffiffiffiffiffiabptan1xffiffiffiffiffiabpa,ðab0Þ61.Zdxaþbx2¼12ffiffiffiffiffiffiffiffiffiabplogaþxffiffiffiffiffiffiffiffiffiabpaxffiffiffiffiffiffiffiffiffiabp,ðab0Þor1ffiffiffiffiffiffiffiffiffiabptanh1xffiffiffiffiffiffiffiffiffiabpa,ðab0Þ8:62.Zdxa2þb2x2¼1abtan1bxa63.Zxdxaþbx2¼12blogðaþbx2Þ64.Zx2dxaþbx2¼xbabZdxaþbx265.Zdxðaþbx2Þ2¼x2aðaþbx2Þþ12aZdxaþbx2A-23INTEGRALS(Continued)66.Zdxa2b2x2¼12ablogaþbxabx67.Zdxðaþbx2Þmþ1¼12maxðaþbx2Þmþ2m12maZdxðaþbx2Þmorð2mÞ!ðm!Þ2x2aXmr¼1r!ðr1Þ!ð4aÞmrð2rÞ!ðaþbx2Þrþ1ð4aÞmZdxaþbx2#8:68.Zxdxðaþbx2Þmþ1¼12bmðaþbx2Þm69.Zx2dxðaþbx2Þmþ1¼x2mbðaþbx2Þmþ12mbZdxðaþbx2Þm70.Zdxxðaþbx2Þ¼12alogx2aþbx271.Zdxx2ðaþbx2Þ¼1axbaZdxaþbx272.Zdxxðaþbx2Þmþ1¼12amðaþbx2Þmþ1aZdxxðaþbx2Þmor12amþ1Xmr¼1arrðaþbx2Þrþlogx2aþbx2#8:73.Zdxx2ðaþbx2Þmþ1¼1aZdxx2ðaþbx2ÞmbaZdxðaþbx2Þmþ174.Zdxaþbx3¼k3a12logðkþxÞ3aþbx3þffiffiffi3ptan12xkkffiffiffi3p ,k¼ffiffiffiab3r75.Zxdxaþbx3¼13bk12logaþbx3ðkþxÞ3þffiffiffi3ptan12xkkffiffiffi3p ,k¼ffiffiffiab3r76.Zx2dxaþbx3¼13blogðaþbx3Þ77.Zdxaþbx4¼k2a12logx2þ2kxþ2k2x22kxþ2k2þtan12kx2k2x2 ,ab0,k¼ffiffiffiffiffia4b4r78.Zdxaþbx4¼k2a12logxþkxkþtan1xk ,ab0,k¼ffiffiffiffiffiffiffiab4r79.Zxdxaþbx4¼12bktan1x2k,ab0,k¼ffiffiffiabr80.Zxdxaþbx4¼14bklogx2kx2þk,ab0,k¼ffiffiffiffiffiffiffiabr81.Zx2dxaþbx4¼14bk12logx22kxþ2k2x2þ2kxþ2k2þtan12kx2k2x2 ,ab0,k¼ffiffiffiffiffia4b4rA-24INTEGRALS(Continued)82.Zx2dxaþbx4¼14bklogxkxþkþ2tan1xk ,ab0,k¼ffiffiffiffiffiffiffiab4r83.Zx3dxaþbx4¼14blogðaþbx4Þ84.ZdxxðaþbxnÞ¼1anlogxnaþbxn85.ZdxðaþbxnÞmþ1¼1aZdxðaþbxnÞmbaZxndxðaþbxnÞmþ186.ZxmdxðaþbxnÞpþ1¼1bZxmndxðaþbxnÞpabZxmndxðaþbxnÞpþ187.ZdxxmðaþbxnÞpþ1¼1aZdxxmðaþbxnÞpbaZdxxmnðaþbxnÞpþ188.ZxmðaþbxnÞpdx¼1bðnpþmþ1Þxmnþ1ðaþbxnÞpþ1aðmnþ1ÞZxmnðaþbxnÞpdx or1npþmþ1xmþ1ðaþbxnÞpþanpZxmðaþbxnÞp1dx or1aðmþ1Þxmþ1ðaþbxnÞpþ1ðmþ1þnpþnÞbZxmþnðaþbxnÞpdx or1anðpþ1Þxmþ1ðaþbxnÞpþ1þðmþ1þnpþnÞZxmðaþbxnÞpþ1dx 8:FORMSCONTAININGc3x389.Zdxc3x3¼16c2logðcxÞ3c3x3þ1c2ffiffiffi3ptan12xccffiffiffi3p90.Zdxðc3x3Þ2¼x3c3ðc3x3Þþ23c3Zdxc3x391.Zdxðc3x3Þnþ1¼13nc3xðc3x3Þnþð3n1ÞZdxðc3x3Þn 92.Zxdxc3x3¼16clogc3x3ðcxÞ31cffiffiffi3ptan12xccffiffiffi3p93.Zxdxðc3x3Þ2¼x23c3ðc3x3Þþ13c3Zxdxc3x3A-25INTEGRALS(Continued)94.Zxdxðc3x3Þnþ1¼13nc3x2ðc3x3Þnþð3n2ÞZxdxðc3x3Þn 95.Zx2dxc3x3¼13logðc3x3Þ96.Zx2dxðc3x3Þnþ1¼13nðc3x3Þn97.Zdxxðc3x3Þ¼13c3logx3c3x398.Zdxxðc3x3Þ2¼13c3ðc3x3Þþ13c6logx3c3x399.Zdxxðc3x3Þnþ1¼13nc3ðc3x3Þnþ1c3Zdxxðc3x3Þn100.Zdxx2ðc3x3