高等数学积分表(同济6版)

(˜)¹kax+bÈ©1.Zdxax+b=1aln|ax+b|+C2.Z(ax+b)μdx=1a(μ+1)(ax+b)μ+1+C(μ6=−1)3.Zxax+bdx=1a2(ax+b−bln|(x+b)|)+C4.Zx2ax+bdx=1a3h12(ax+b)2−2b(ax+b)+b2ln|ax+b|i+C5.Zdxx(ax+b)=−1blnax+bx+C6.Zdxx2(ax+b)=−1bx+ab2lnax+bx+C7.Zx(ax+b)2dx=1a2ln|ax+b|+bax+b+C8.Zx2(ax+b)2dx=1a3ax+b−2bln|ax+b|−b2ax+b+C9.Zdxx(a+b)2=1b(ax+b)−1b2lnax+bx+C()¹k√ax+bÈ©10.Z√ax+bdx=23aq(ax+b)3+C11.Zx√ax+bdx=215a2(3ax−2b)q(ax+b)3+C12.Zx2√ax+bdx=2105a3(15a2x2−12abx+8b2)q(ax+b)3+C13.Zx√ax+bdx=23a2(ax−2b)√ax+b+C14.Zx2√ax+bdx=215a2(2a2x2−4abx+8b2)√ax+b+C115.Zdxx√ax+b=1√bln√ax+b−√b√ax+b+√b+C(b0)2−√−barctanqax+b−b+C(b0)16.Zdxx2√ax+b=−√ax+bbx−a2bZdxx√ax+b17.Z√ax+bxdx=2√ax+b+bZdxx√ax+b18.Z√ax+bx2dx=−√ax+bx+a2Zdxx√ax+b(n)¹kx2±aÈ©19.Zdxx2+a2=1aarctanxa+C20.Zdx(x2+a2)n=x2(n−1)a2(x2+a2)n−1+2n−32(n−1)a2Zdx(x2+a2)n−121.Zdxx2−a2=12alnx−ax+a+C(o)¹kax2+bÈ©22.Zdxax2+b=1√abarctanqabx+C(b0)12√−abln√ax−√−b√ax+√−b+C(b0)23.Zxax2+bdx=12aln|ax2+b|+C24.Zx2ax2+bdx=xa−baZdxax2+b25.Zdxx(ax2+b)=12blnx2|ax2+b|+C26.Zdxx2(ax2+b)=−1bx−abZdxax2+b27.Zdxx3(ax2+b)=a2b2ln|ax2+b|x2−12bx2+C228.Zdx(ax2+b)2=x2b(ax2+b)+12bZdxax2+b(Ê)¹kax2+bx+cÈ©29.Zdxax2+bx+cdx=2√4ac−b2arctan2ax+b√4ac−b2+C(b24ac)1√b2−4acln2ax+b−√b2−4ac2ax+b+√b2−4ac+C(b24ac)30.Zxax2+bx+cdx=12aln|ax2+bx+c|−b2aZdxax2+bx+c(8)¹k√x2+a2(a0)È©31.Zdx√x2+a2=arshxa+C1=ln(x+√x2+a2)+C32.Zdxq(x2+a2)3=xa2√x2+a2+C33.Zx√x2+a2dx=√x2+a2+C34.Zxq(x2+a2)3dx=−1√x2+a2+C35.Zx2√x2+a2dx=x2√x2+a2−a22ln(x+√x2+a2)+C36.Zx2q(x2+a2)3dx=−x√x2+a2+ln(x+√x2+a2)+C37.Zdxx√x2+a2=1aln√x2+a2−a|x|+C38.Zdxx2√x2+a2=−√x2+a2a2x+C39.Z√x2+a2dx=x2√x2+a2+a22ln(x+√x2+a2)+C40.Zq(x2+a2)3dx=x8(2x2+5a2)√x2+a2+38a4ln(x+√x2+a2)+C341.Zx√x2+a2dx=13q(x2+a2)3+C42.Zx2√x2+a2dx=x8(2x2+a2)√x2+a2−a48ln(x+√x2+a2)+C43.Z√x2+a2xdx=√x2+a2+aln√x2+a2−a|x|+C44.Z√x2+a2x2dx=−√x2+a2x+ln(x+√x2+a2)+C(Ô)¹k√x2−a2(a0)È©45.Zdx√x2−a2=x|x|arch|x|a+C1=ln|x+√x2−a2|+C46.Zdxq(x2−a2)3=−xa2√x2−a2+C47.Zx√x2−a2dx=√x2−a2+C48.Zxq(x2−a2)3dx=−1√x2−a2+C49.Zx2√x2−a2dx=x2√x2−a2+a22ln|x+√x2−a2|+C50.Zx2q(x2−a2)3dx=−x√x2−a2+ln|x+√x2−a2|+C51.Zdxx√x2−a2=1aarccosa|x|+C52.Zdxx2√x2−a2=√x2−a2a2x+C53.Z√x2−a2dx=x2√x2−a2−a22ln|x+√x2−a2|+C54.Zq(x2−a2)3dx=x8(2x2−5a2)√x2−a2+38a4ln|x+√x2−a2|+C455.Zx√x2−a2dx=13q(x2−a2)3+C56.Zx2√x2−a2dx=x8(2x2−a2)√x2−a2−a48ln|x+√x2−a2|+C57.Z√x2−a2xdx=√x2−a2−arccosa|x|+C58.Z√x2−a2x2dx=−√x2−a2x+ln|x+√x2−a2|+C(l)¹k√a2−x2(a0)È©59.Zdx√a2−x2=arcsinxa+C60.Zdxq(a2−x2)3=xa2√a2−x2+C61.Zx√a2−x2dx=−√a2−x2+C62.Zxq(a2−x2)3dx=−1√a2−x2+C63.Zx2√a2−x2dx=−x2√a2−x2+a22arcsinxa+C64.Zx2q(a2−x2)3dx=x√a2−x2−arcsinxa+C65.Zdxx√a2−x2=1alna−√a2−x2|x|+C66.Zdxx2√a2−x2=−a2−x2a2x+C67.Z√a2−x2dx=x2√a2−x2+a22arcsinxa+C68.Zq(a2−x2)3dx=x8(5a2−2x2)√a2−x2+38a4arcsinxa+C569.Zx√a2−x2dx=−13q(a2−x2)3+C70.Zx2√a2−x2dx=x8(2x2−a2)√a2−x2+a48arcsinxa+C71.Z√a2−x2xdx=√a2−x2+alna−√a2−x2|x|+C72.Z√a2−x2x2dx=−√a2−x2x−arcsinxa+C(Ê)¹k√±ax2+bx+c(a0)È©73.Zdx√ax2+bx+c=1√aln|2ax+b+2√a√ax2+bx+c|+C74.Z√ax2+bx+cdx=2ax+b4a√ax2+bx+c+4ac−b28√a3ln|2ax+b+2√a√ax2+bx+c|+C75.Zx√ax2+bx+cdx=1a√ax2+bx+c−b2√a3ln|2ax+b+2√a√ax2+bx+c|+C76.Zdx√c+bx−ax2=−1√aarcsin2ax−b√b2+4ac+C77.Z√c+bx−ax2dx=2ax−b4a√c+bx−ax2+b2+4ac8√a3arcsin2ax−b√b2+4ac+C78.Zx√c+bx−ax2dx=−1a√c+bx−ax2+b2√a3arcsin2ax−b√b2+4ac+C(›)¹kq±x−ax+a½öq(x−a)(b−x)È©79.Zsx−ax−bdx=(x−b)sx−ax−b+(b−a)ln(q|x−a|+q|x−b|)+C680.Zsx−ax−bdx=(x−b)sx−ax−b+(b−a)arcsinsx−ab−a+C81.Zdxq(x−a)(x−b)=2arcsinsx−ab−a+C(ab)82.Zq(x−a)(b−x)dx=2x−a−b4q(x−a)(b−x)+(b−a)24arcsinsx−ab−a+C(ab)(›˜)¹kn¼ê¼êÈ©83.Zsinxdx=−cosx+C84.Zcosxdx=sinx+C85.Ztanxdx−ln|cosx|+C86.Zctgxdx=ln|sinx|+C87.Zsecxdx=ln|tan(π4+x2)|+C=ln|secx+tanx|+C88.Zcscxdx=ln|tanx2|+C=ln|cscx−ctgx|+C89.Zsec2xdx=tanx+C90.Zcsc2xdx=−ctgx+C91.Zsecxtanxdx=secx+C92.Zcscxdxctgxdx=−cscx+C93.Zsin2xdx=x2−14sin2x+C794.Zcos2xdx=x2+14sin2x+C95.Zsinnxdx=−1nsinn−1xcosx+n−1nZsinn−2dx96.Zcosnxdx=1ncosn−1xsinx+n−1nZcosn−2xdx97.Zdxsinnx=−1n−1.cosxsinn−1x+n−2n−1Zdxsinn−2x98.Zdxcosnx=1n−1.sinxcosn−1x+n−2n−1Zdxcosxn−2x99.Zcosmsinnxdx=1m+ncosm−1xsinn+1x+m−1m+nZcosm−2xsinnxdx=−1m+1cosm+1xsinn−1x+n−1m+nZcosmxsinn−2xdx100.Zsinaxcosbxdx=−12(a+b)cos(a+b)x−12(a−b)cos(a−b)x+C101.Zsinaxsinbxdx=−12(a+b)sin(a+b)x+12(a−b)sin(a−b)x+C102.Zcosaxcosbxdx=12(a+b)sin(a+b)x+12(a−b)sin(a−b)x+C103.Zdxa+bsinx=2√a2−b2arctanarctanx2+b√a2−b2+C(a2b2)104.Zdxa+bsinx=1√b2−a2lnarctanx2+b−√b2−a2arctanx2+b+√b2−a2+C(a2b2)105.Zdxa+bcosx=2a+bsa+ba−barctansa−ba+btanx2!+C(a2b2)106.Zdxa+bcosx=1a+bsa+bb−alntanx2+qa+bb−atanx2−qa+bb−a+C(a2b2)107.Zdxa2cos2x+b2sin2x=1abarctan(batanx)+C8108.Zdxa2cos2x−b2sin2x=12ablnbtanx+abtanx−a+C109.Zxsinaxdx=1a2sinax−1axcosax+C110.Zx2sinaxdx=−1ax2cosax+2a2xsinax+2a3cosax+C111.Zxcosaxdx=1a2cosax+1axsinax+C112.Zx2cosaxdx=1ax2sinax+2a2xcosax−2a3sinax+C(›)¹k‡n¼êÈ©(Ù¥a0)113.Zarcsinxadx=xarcsinxa+√a2−x2+C114.Zxarcsinxadx=x22−a24arcsinxa+x4√a2−x2+C115.Zx2arcsinxadx=x33arcsinxa+19(x2+2a2)√a2−x2+C116.Zarccosxadx=xarccosxa−√a2−x2+C117.Zxarccosxadx=x22−a24arccosx4−x4√a2−x2+C118.Zx2arccosxadx=x3aarccosxa−19(x2+2a2)√a2−x2+C119.Zarctanxadx=xarctanxa−a2ln(a2+x2)+C120.Zxarctanxadx=12(a2+x2)arctanxa−a2x+C121.Zx2arctanxadx=x33arctanxa−a6x