Table of Basic Integrals Basic Forms

1 Table of Basic IntegralsBasic Forms (1) xndx=1n+ 1xn+1, n6= 1(2) 1xdx= ln|x|(3) udv=uv vdu(4) 1ax+bdx=1aln|ax+b| Integrals of Rational Functions(5) 1(x+a)2dx= 1x+a(6) (x+a)ndx=(x+a)n+1n+ 1,n6= 1(7) x(x+a)ndx=(x+a)n+1((n+ 1)x a)(n+ 1)(n+ 2)(8) 11 +x2dx= tan 1x(9) 1a2+x2dx=1atan 1xa1(10) xa2+x2dx=12ln|a2+x2|(11) x2a2+x2dx=x atan 1xa(12) x3a2+x2dx=12x2 12a2ln|a2+x2|(13) 1ax2+bx+cdx=2 4ac b2tan 12ax+b 4ac b2(14) 1(x+a)(x+b)dx=1b alna+xb+x, a6=b(15) x(x+a)2dx=aa+x+ ln|a+x|(16) xax2+bx+cdx=12aln|ax2+bx+c| ba 4ac b2tan 12ax+b 4ac b2 Integrals with Roots(17) x a dx=23(x a)3/2(18) 1 x adx= 2 x a(19) 1 a xdx= 2 a x2(20) x x a dx= 2a3(x a)3/2+25(x a)5/2,or23x(x a)3/2 415(x a)5/2,or215(2a+ 3x)(x a)3/2(21) ax+b dx=(2b3a+2x3) ax+b(22) (ax+b)3/2dx=25a(ax+b)5/2(23) x x adx=23(x 2a) x a(24) xa xdx= x(a x) atan 1 x(a x)x a(25) xa+xdx= x(a+x) aln[ x+ x+a](26) x ax+b dx=215a2( 2b2+abx+ 3a2x2) ax+b(27) x(ax+b)dx=14a3/2[(2ax+b) ax(ax+b) b2ln a x+ a(ax+b) ](28) x3(ax+b)dx=[b12a b28a2x+x3] x3(ax+b)+b38a5/2ln a x+ a(ax+b) (29) x2 a2dx=12x x2 a2 12a2ln x+ x2 a2 3(30) a2 x2dx=12x a2 x2+12a2tan 1x a2 x2(31) x x2 a2dx=13(x2 a2)3/2(32) 1 x2 a2dx= ln x+ x2 a2 (33) 1 a2 x2dx= sin 1xa(34) x x2 a2dx= x2 a2(35) x a2 x2dx= a2 x2(36) x2 x2 a2dx=12x x2 a2 12a2ln x+ x2 a2 (37) ax2+bx+c dx=b+ 2ax4a ax2+bx+c+4ac b28a3/2ln 2ax+b+ 2 a(ax2+bx+c) x ax2+bx+c dx=148a5/2(2 a ax2+bx+c( 3b2+ 2abx+ 8a(c+ax2))+3(b3 4abc) ln b+ 2ax+ 2 a ax2+bx+c )(38)4(39) 1 ax2+bx+cdx=1 aln 2ax+b+ 2 a(ax2+bx+c) (40) x ax2+bx+cdx=1a ax2+bx+c b2a3/2ln 2ax+b+ 2 a(ax2+bx+c) (41) dx(a2+x2)3/2=xa2 a2+x2 Integrals with Logarithms(42)

2 Lnax dx=xlnax x(43) xlnx dx=12x2lnx x24(44) x2lnx dx=13x3lnx x39(45) xnlnx dx=xn+1(lnxn+ 1 1(n+ 1)2), n6= 1(46) lnaxxdx=12(lnax)2(47) lnxx2dx= 1x lnxx5(48) ln(ax+b)dx=(x+ba)ln(ax+b) x,a6= 0(49) ln(x2+a2)dx=xln(x2+a2) + 2atan 1xa 2x(50) ln(x2 a2)dx=xln(x2 a2) +alnx+ax a 2x(51) ln(ax2+bx+c)dx=1a 4ac b2tan 12ax+b 4ac b2 2x+(b2a+x)ln(ax2+bx+c)(52) xln(ax+b)dx=bx2a 14x2+12(x2 b2a2)ln(ax+b)(53) xln(a2 b2x2)dx= 12x2+12(x2 a2b2)ln(a2 b2x2)(54) (lnx)2dx= 2x 2xlnx+x(lnx)2(55) (lnx)3dx= 6x+x(lnx)3 3x(lnx)2+ 6xlnx(56) x(lnx)2dx=x24+12x2(lnx)2 12x2lnx(57) x2(lnx)2dx=2x327+13x3(lnx)2 29x3lnx6 Integrals with Exponentials(58) eaxdx=1aeax(59) xeaxdx=1a xeax+i 2a3/2erf(i ax),where erf(x) =2 x0e t2dt(60) xexdx= (x 1)ex(61) xeaxdx=(xa 1a2)eax(62) x2exdx=(x2 2x+ 2)ex(63) x2eaxdx=(x2a 2xa2+2a3)eax(64) x3exdx=(x3 3x2+ 6x 6)ex(65) xneaxdx=xneaxa na xn 1eaxdx(66) xneaxdx=( 1)nan+1 [1 +n, ax],where (a,x) = xta 1e tdt(67) eax2dx= i 2 aerf(ix a)7(68) e ax2dx= 2 aerf(x a)(69) xe ax2dx= 12ae ax2(70) x2e ax2dx=14 a3erf(x a) x2ae ax2 Integrals with Trigonometric Functions(71) sinax dx= 1acosax(72) sin2ax dx=x2 sin 2ax4a(73) sin3ax dx= 3 cosax4a+cos 3ax12a(74) sinnax dx= 1acosax2F1[12,1 n2,32,cos2ax](75) cosax dx=1asinax(76) cos2ax dx=x2+sin 2ax4a(77) cos3axdx=3 sinax4a+sin 3ax12a8(78) cospaxdx= 1a(1 +p)cos1+pax 2F1[1 +p2,12,3 +p2,cos2ax](79) cosxsinx dx=12sin2x+c1= 12cos2x+c2= 14cos 2x+c3(80) cosaxsinbx dx=cos[(a b)x]2(a b) cos[(a+b)x]2(a+b),a6=b(81) sin2axcosbx dx= sin[(2a b)x]4(2a b)+sinbx2b sin[(2a+b)x]4(2a+b)(82) sin2xcosx dx=13sin3x(83) cos2axsinbx dx=cos[(2a b)x]4(2a b) cosbx2b cos[(2a+b)x]4(2a+b)

3 (84) cos2axsinax dx= 13acos3ax(85) sin2axcos2bxdx=x4 sin 2ax8a sin[2(a b)x]16(a b)+sin 2bx8b sin[2(a+b)x]16(a+b)(86) sin2axcos2ax dx=x8 sin 4ax32a(87) tanax dx= 1aln cosax9(88) tan2ax dx= x+1atanax(89) tannax dx=tann+1axa(1 +n) 2F1(n+ 12,1,n+ 32, tan2ax)(90) tan3axdx=1aln cosax+12asec2ax(91) secx dx= ln|secx+ tanx|= 2 tanh 1(tanx2)(92) sec2ax dx=1atanax(93) sec3x dx=12secxtanx+12ln|secx+ tanx|(94) secxtanx dx= secx(95) sec2xtanx dx=12sec2x(96) secnxtanx dx=1nsecnx,n6= 0(97) cscx dx= ln tanx2 = ln|cscx cotx|+C10(98) csc2ax dx= 1acotax(99) csc3x dx= 12cotxcscx+12ln|cscx cotx|(100) cscnxcotx dx= 1ncscnx,n6= 0(101) secxcscx dx= ln|tanx|Products of Trigonometric Functions and Mono-mials(102) xcosx dx= cosx+xsinx(103) xcosax dx=1a2cosax+xasinax(104) x2cosx dx= 2xcosx+(x2 2)sinx(105) x2cosax dx=2xcosaxa2+a2x2 2a3sinax(106) xncosxdx= 12(i)n+1[ (n+ 1, ix) + ( 1)n (n+ 1,ix)]11(107) xncosax dx=12(ia)1 n[( 1)n (n+ 1, iax) (n+ 1,ixa)](108) xsinx dx= xcosx+ sinx(109) xsinax dx= xcosaxa+sinaxa2(110) x2sinx dx=(2 x2)cosx+ 2xsinx(111) x2sinax dx=2 a2x2a3cosax+2xsinaxa2(112) xnsinx dx= 12(i)n[ (n+ 1, ix) ( 1)n (n+ 1, ix)](113) xcos2x dx=x24+18cos 2x+14xsin 2x(114) xsin2x dx=x24 18cos 2x 14xsin 2x(115) xtan2x dx= x22+ ln cosx+xtanx(116) xsec2x dx= ln cosx+xtanx12 Products of Trigonometric Functions and Ex-ponentials(117) exsinx dx=12ex(sinx cosx)(118) ebxsinax dx=1a2+b2ebx(bsinax acosax)(119) excosx dx=12ex(sinx+ cosx)(120) ebxcosax dx=1a2+b2ebx(asinax+bcosax)(121) xexsinx dx=12ex(cosx xcosx+xsinx)(122) xexcosx dx=12ex(xcosx sinx+xsinx) Integrals of Hyperbolic Functions(123)

4 Coshax dx=1asinhax(124) eaxcoshbx dx= eaxa2 b2[acoshbx bsinhbx]a6=be2ax4a+x2a=b(125) sinhax dx=1acoshax13(126) eaxsinhbx dx= eaxa2 b2[ bcoshbx+asinhbx]a6=be2ax4a x2a=b(127) tanhaxdx=1aln coshax(128) eaxtanhbx dx= e(a+2b)x(a+ 2b)2F1[1 +a2b,1,2 +a2b, e2bx] 1aeax2F1[1,a2b,1 +a2b, e2bx]a6=beax 2 tan 1[eax]aa=b(129) cosaxcoshbx dx=1a2+b2[asinaxcoshbx+bcosaxsinhbx](130 ) cosaxsinhbx dx=1a2+b2[bcosaxcoshbx+asinaxsinhbx](131 ) sinaxcoshbx dx=1a2+b2[ acosaxcoshbx+bsinaxsinhbx](132) sinaxsinhbx dx=1a2+b2[bcoshbxsinax acosaxsinhbx](133) sinhaxcoshaxdx=14a[ 2ax+ sinh 2ax](134) sinhaxcoshbx dx=1b2 a2[bcoshbxsinhax acoshaxsinhbx]c 2014. , last revised June 14, 2014. This mate-rial is provided as is without warranty or representation about the accuracy, correctness orsuitability of this material for any purpose. This work is licensed under the Creative Com-mons Attribution-Noncommercial-Share Alike United States License.

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