Table of Integrals

1 Table of Integrals Basic Forms xndx=1n+ 1xn+1(1) 1xdx= ln|x|(2) udv=uv vdu(3) 1ax+bdx=1aln|ax+b|(4) Integrals of Rational Functions 1(x+a)2dx= 1x+a(5) (x+a)ndx=(x+a)n+1n+ 1,n6= 1(6) x(x+a)ndx=(x+a)n+1((n+ 1)x a)(n+ 1)(n+ 2)(7) 11 +x2dx= tan 1x(8) 1a2+x2dx=1atan 1xa(9) xa2+x2dx=12ln|a2+x2|(10) x2a2+x2dx=x atan 1xa(11) x3a2+x2dx=12x2 12a2ln|a2+x2|(12) 1ax2+bx+cdx=2 4ac b2tan 12ax+b 4ac b2(13) 1(x+a)(x+b)dx=1b alna+xb+x, a6=b(14) x(x+a)2dx=aa+x+ ln|a+x|(15) xax2+bx+cdx=12aln|ax2+bx+c| ba 4ac b2tan 12ax+b 4ac b2(16) Integrals with Roots x adx=23(x a)3/2(17) 1 x adx= 2 x a(18) 1 a xdx= 2 a x(19) x x adx=23a(x a)3/2+25(x a)5/2(20) ax+bdx=(2b3a+2x3) ax+b(21) (ax+b)3/2dx=25a(ax+b)5/2(22) x x adx=23(x 2a) x a(23) xa xdx= x(a x) atan 1 x(a x)x a(24) xa+xdx= x(a+x) aln[ x+ x+a](25) x ax+bdx=215a2( 2b2+abx+ 3a2x2) ax+b(26) x(ax+b)dx=14a3/2[(2ax+b) ax(ax+b) b2ln a x+ a(ax+b) ](27) x3(ax+b)dx=[b12a b28a2x+x3] x3(ax+b)+b38a5/2ln a x+ a(ax+b) (28) x2 a2dx=12x x2 a2 12a2ln x+ x2 a2 (29) a2 x2dx=12x a2 x2+12a2tan 1x a2 x2(30) x x2 a2dx=13(x2 a2)3/2(31) 1 x2 a2dx= ln x+ x2 a2 (32) 1 a2 x2dx= sin 1xa(33) x x2 a2dx= x2 a2(34) x a2 x2dx= a2 x2(35) x2 x2 a2dx=12x x2 a2 12a2ln x+ x2 a2 (36) ax2+bx+cdx=b+ 2ax4a ax2+bx+c+4ac b28a3/2ln 2ax+b+ 2 a(ax2+bx+c) (37) x ax2+bx+c=148a5/2(2 a ax2+bx+c ( 3b2+ 2abx+ 8a(c+ax2))+3(b3 4abc) ln b+ 2ax+ 2 a ax2+bx+c )(38) 1 ax2+bx+cdx=1 aln 2ax+b+ 2 a(ax2+bx+c) (39) x ax2+bx+cdx=1a ax2+bx+c b2a3/2ln 2ax+b+ 2 a(ax2+bx+c) (40) dx(a2+x2)3/2=xa2 a2+x2(41) Integrals with Logarithms lnaxdx=xlnax x(42) lnaxxdx=12(lnax)2(43) ln(ax+b)dx=(x+ba)ln(ax+b) x,a6= 0(44) ln(x2+a2) dx =xln(x2+a2) + 2atan 1xa 2x(45) ln(x2 a2) dx =xln(x2 a2) +alnx+ax a 2x(46) ln(ax2+bx+c)dx=1a 4ac b2tan 12ax+b 4ac b2 2x+(b2a+x)ln(ax2+bx+c)(47) xln(ax+b)dx=bx2a 14x2+12(x2 b2a2)ln(ax+b)(48) xln(a2 b2x2)dx= 12x2+12(x2 a2b2)

2 Ln(a2 b2x2)(49) Integrals with Exponentials eaxdx=1aeax(50) xeaxdx=1a xeax+i 2a3/2erf(i ax),where erf(x) =2 x0e t2dt(51) xexdx= (x 1)ex(52) xeaxdx=(xa 1a2)eax(53) x2exdx=(x2 2x+ 2)ex(54) x2eaxdx=(x2a 2xa2+2a3)eax(55) x3exdx=(x3 3x2+ 6x 6)ex(56) xneaxdx=xneaxa na xn 1eaxdx(57) xneaxdx=( 1)nan+1 [1 +n, ax],where (a,x) = xta 1e tdt(58) eax2dx= i 2 aerf(ix a)(59) e ax2dx= 2 aerf(x a)(60) xe ax2dx = 12ae ax2(61) x2e ax2dx =14 a3erf(x a) x2ae ax2(62) 2014. , last revised June 14, 2014. This material is provided as is without warranty or representation about the accuracy, correctness orsuitability of the material for any purpose, and is licensed under the Creative Commons Attribution-Noncommercial-ShareAlike United States License. To view a copy of thislicense, send a letter to Creative Commons, 171 Second Street, Suite 300, San Francisco, California, 94105, with Trigonometric Functions sinaxdx= 1acosax(63) sin2axdx=x2 sin 2ax4a(64) sinnaxdx= 1acosax2F1[12,1 n2,32,cos2ax](65) sin3axdx= 3 cosax4a+cos 3ax12a(66) cosaxdx=1asinax(67) cos2axdx=x2+sin 2ax4a(68) cospaxdx= 1a(1 +p)cos1+pax 2F1[1 +p2,12,3 +p2,cos2ax](69) cos3axdx=3 sinax4a+sin 3ax12a(70) cosaxsinbxdx=cos[(a b)x]2(a b) cos[(a+b)x]2(a+b),a6=b(71) sin2axcosbxdx= sin[(2a b)x]4(2a b)+sinbx2b sin[(2a+b)x]4(2a+b)(72) sin2xcosxdx=13sin3x(73) cos2axsinbxdx=cos[(2a b)x]4(2a b) cosbx2b cos[(2a+b)x]4(2a+b)(74) cos2axsinaxdx= 13acos3ax(75) sin2axcos2bxdx=x4 sin 2ax8a sin[2(a b)x]16(a b)+sin 2bx8b sin[2(a+b)x]16(a+b)(76) sin2axcos2axdx=x8 sin 4ax32a(77) tanaxdx= 1aln cosax(78) tan2axdx= x+1atanax(79) tannaxdx=tann+1axa(1 +n) 2F1(n+ 12,1,n+ 32, tan2ax)(80) tan3axdx=1aln cosax+12asec2ax(81)

3 Secxdx= ln|secx+ tanx|= 2 tanh 1(tanx2)(82) sec2axdx=1atanax(83) sec3xdx =12secxtanx+12ln|secx+ tanx|(84) secxtanxdx= secx(85) sec2xtanxdx=12sec2x(86) secnxtanxdx=1nsecnx,n6= 0(87) cscxdx= ln tanx2 = ln|cscx cotx|+C(88) csc2axdx= 1acotax(89) csc3xdx= 12cotxcscx+12ln|cscx cotx|(90) cscnxcotxdx= 1ncscnx,n6= 0(91) secxcscxdx= ln|tanx|(92)Products of Trigonometric Functions andMonomials xcosxdx= cosx+xsinx(93) xcosaxdx=1a2cosax+xasinax(94) x2cosxdx= 2xcosx+(x2 2)sinx(95) x2cosaxdx=2xcosaxa2+a2x2 2a3sinax(96) xncosxdx= 12(i)n+1[ (n+ 1, ix)+( 1)n (n+ 1,ix)](97) xncosaxdx=12(ia)1 n[( 1)n (n+ 1, iax) (n+ 1,ixa)](98) xsinxdx= xcosx+ sinx(99) xsinaxdx= xcosaxa+sinaxa2(100) x2sinxdx=(2 x2)cosx+ 2xsinx(101) x2sinaxdx=2 a2x2a3cosax+2xsinaxa2(102) xnsinxdx= 12(i)n[ (n+ 1, ix) ( 1)n (n+ 1, ix)](103)Products of Trigonometric Functions andExponentials exsinxdx=12ex(sinx cosx)(104) ebxsinaxdx=1a2+b2ebx(bsinax acosax)(105) excosxdx=12ex(sinx+ cosx)(106) ebxcosaxdx=1a2+b2ebx(asinax+bcosax)(107) xexsinxdx=12ex(cosx xcosx+xsinx)(108) xexcosxdx=12ex(xcosx sinx+xsinx)(109) Integrals of Hyperbolic Functions coshaxdx=1asinhax(110) eaxcoshbxdx= eaxa2 b2[acoshbx bsinhbx]a6=be2ax4a+x2a=b(111) sinhaxdx=1acoshax(112) eaxsinhbxdx= eaxa2 b2[ bcoshbx+asinhbx]a6=be2ax4a x2a=b(113) eaxtanhbxdx= e(a+2b)x(a+ 2b)2F1[1 +a2b,1,2 +a2b, e2bx] 1aeax2F1[a2b,1,1E, e2bx]a6=beax 2 tan 1[eax]aa=b(114) tanhaxdx=1aln coshax(115) cosaxcoshbxdx=1a2+b2[asinaxcoshbx+bcosax sinhbx](116) cosaxsinhbxdx=1a2+b2[bcosaxcoshbx+asinax sinhbx](117) sinaxcoshbxdx=1a2+b2[ acosaxcoshbx+bsinaxsinhbx](118) sinaxsinhbxdx=1a2+b2[bcoshbxsinax acosaxsinhbx](119) sinhaxcoshaxdx=14a[ 2ax+ sinh 2ax](120) sinhaxcoshbxdx=1b2 a2[bcoshbxsinhax acoshaxsinhbx](121)2