Integration Formulas - mathportal.org

1 Integration Formulas 1. Common Integrals Indefinite Integral Method of substitution ( ( )) ( )( )f g x g x dxf u du = Integration by parts ( ) ( )( ) ( )( ) ( )f x g x dxf x g xg x f x dx = Integrals of Rational and Irrational Functions 11nnxx dxCn+=++ 1lndxx Cx=+ c dx cx C=+ 22xxdxC=+ 323xx dxC=+ 211dxCxx= + 23x xxdxC=+ 21arctan1dxx Cx=++ 21arcsin1dxx Cx=+ Integrals of Trigonometric Functions sincosx dxx C= + cossinx dxx C=+ tanln secx dxx C=+ secln tansecx dxxx C=++ ()21sinsin cos2x dxxxxC= + ()21cossin cos2x dxxxxC=++ 2tantanx dxx x C= + 2sectanx dxx C=+ Integrals of Exponential and Logarithmic Functions lnlnx dx x x x C= + ()112lnln11nnnxxxx dxxCnn++= +++ xxe dx eC=+ lnxxbb dxCb=+ sinhcoshx dxx C=+ coshsinhx dxx C=+ 2.

2 Integrals of Rational Functions Integrals involving ax + b ()()()()111nnax bax b dxafo nnr+++=+ 11lndxax bax ba=++ ()()() ()()()12112,12nna nx bx ax b dxax ba nnfor nn+ + +=+++ 2lnxxbdxax bax baa= ++ ()()2221lnxbdxax ba ax baax b=++++ ()()() () ()()12121,21nnan x bxdxax ba nnfor nax bn =+ + ()()222312ln2ax bxdxb ax bbax bax ba + = +++ + ()222312 lnxbdxax bbax bax baax b =+ + ++ ()()2233212ln2xbbdxax bax baax bax b =++ +++ ()()()()()321223213211, 2, 3nnnnax bb a bb ax bxdxnnfonar nax b +++ = + + ()11lnax bdxx ax bbx+= + ()2211lnaax bdxbxxx ax bb+= ++ ()()222321112lnax bdxaxb a xbab x bx ax b += + ++ Integrals involving ax2 + bx + c 2211xdxarctgaaxa=+ 221ln121ln2a xfor xaaa xdxx axafor xaax a < += > + 22222222222arctan40441224ln404242402ax bfor ac bac bac bax bbacdxfor ac baxbx cbacax bbacfor ac bax b+ > + = < ++ + + = + 2221ln22xbdxdxaxbx caaaxbx caxbx c=++ ++++ ()

3 2222222222222lnarctan4024422lnarctanh402 442ln4022man bmax baxbx cfor ac baaac bac bmx nman bmax bdxaxbx cfor ac baaxbx ca bacbacman bmaxbx cfor ac baa ax b ++++ > + + =+++ < ++ ++ =+ ()()() ()()()()()112222223 21211 41 4nnnnaax bdxdxnac baxbx cnac baxbx caxbx c +=+ ++ ++++ ()2222111ln22xbdxdxccaxbx caxbx cx axbx c= ++++++ 3. Integrals of Exponential Functions ()21cxcxexe dxcxc= 222322cxcxxxx e dx eccc = + 11n cxn cxncxnx e dxx ex e dxcc = ()1ln!icxicxedxxxi i ==+ ()1lnlncxcxiexdxexE cxc=+ ()22sinsincoscxcxeebxdxcbx bbxcb= + ()22coscossincxcxeebxdxcbx bbxcb=++ ()()1222221sinsinsincossincxncxncxnn nexexdxcx nbxedxcncn = +++ 4.

4 Integrals of Logarithmic Functions lnlncxdxx cx x= ln()ln()ln()bax b dxxax bxax ba+=+ ++ ()()22lnln2 ln2x dxxxx xx= + ()()()1lnlnlnnnncx dxxcxncxdx = ()2lnln lnlnln!inxdxxxxi i ==++ ()() ()()()11111ln1 lnlnnnnfor ndxxdxnxnxx = + ()()12ln1n11l1mmxxxdxxmmfor m+ = ++ ()()()()11lnln111lnnmnnmmxxnxx dxxxdxmrmfo m+ = ++ ()()()1lnln11nnxxdxfor nxn+= + ()()2lnln02nnxxdxfor nxn= ()()()121lnln1111mmmxxdxxmxmforxm = ()()()()()11lnlnn1l11nnnmmmxxxndxdxmxmxx for m = + ln lnlndxxxx= ()() ()11lnln ln1!lniiininxdxxi ixx = =+ ()() ()()11ln1 ln1nndxxxnfxor n = ()()22221lnln22 tanxxa dxxxaxaa +=+ + ()()()()sin lnsin lncos ln2xx dxxx= ()()()()cos lnsin lncos ln2xx dxxx=+ 5.

5 Integrals of Trig. Functions sincosxdxx= cossinxdxx= 21sinsin 224xxdxx= 21cossin 224xxdxx=+ 331sincoscos3xdxxx= 331cossinsin3xdxxx= ln tansin2dxxxdxx= ln tancos24dxxxdxx =+ 2cotsindxxdxxx= 2tancosdxxdxxx= 32cos1ln tansin2 sin22dxxxxx= + 32sin1ln tan224cos2 cosdxxxxx =++ 1sin coscos 24xxdxx= 231sincossin3xxdxx= 231sin coscos3xxdxx= 221sincossin 4832xxxdxx= tanln cosxdxx= 2sin1coscosxdxxx= 2sinln tansincos24xxdxxx =+ 2tantanxdxx x= cotln sinxdxx= 2cos1sinsinxdxxx= 2cosln tancossin2xxdxxx=+ 2cotcotxdxx x= ln tansin cosdxxxx= 21ln tansin24sincosdxxxxx =

6 ++ 21ln tancos2sin cosdxxxxx=+ 22tancotsincosdxxxxx= ()()()()22sinsinsinsin22m n xm n xmxnxdxnm nmnm+ ++ = ()()()()22coscossincos22m n xm n xmxnxdxnm nmnm+ + = ()()()()22sinsincoscos22m n xm n xmxnxdxm nm nmn+ =++ 1cossin cos1nnxxxdxn+= + 1sinsincos1nnxxxdxn+=+ 2arcsinarcsin1xdxxxx=+ 2arccosarccos1xdxxxx= ()21arctanarctanln12xdxxxx= + ()21arc cotarc cotln12xdxxxx=++