Transcription of Table of Integrals - UMD
1 2005 BE Shapiro Page 1 This document may not be reproduced, posted or published without permission. The copyright holder makes no representation about the accuracy, correctness, or suitability of this material for any purpose. Table of Integrals BASIC FORMS (1) xndx!=1n+1xn+1 (2) 1xdx!=lnx (3) udv!=uv"vdu! (4) u(x)!v(x)dx"=u(x)v(x)#v(x)!u(x)dx" RATIONAL FUNCTIONS (5) 1ax+bdx!=1aln(ax+b) (6) 1(x+a)2dx!="1x+a (7) (x+a)ndx!=(x+a)na1+n+x1+n"#$%&', n!"1 (8) x(x+a)ndx!=(x+a)1+n(nx+x"a)(n+2)(n+1) (9) dx1+x2!=tan"1x (10) dxa2+x2!=1atan"1(x/a) (11) xdxa2+x2!=12ln(a2+x2) (12) x2dxa2+x2!=x"atan"1(x/a) (13) x3dxa2+x2!=12x2"12a2ln(a2+x2) (14) (ax2+bx+c)!
2 1dx"=24ac!b2tan!12ax+b4ac!b2#$%&'( (15) 1(x+a)(x+b)dx=!1b"aln(a+x)"ln(b+x)[], a!b (16) x(x+a)2dx=!aa+x+ln(a+x) (17) xax2+bx+cdx!=ln(ax2+bx+c)2a!!!!!"ba4ac"b 2tan"12ax+b4ac"b2#$%&'( Integrals WITH ROOTS (18) x!adx"=23(x!a)3/2 (19) 1x adx!=2x a (20) 1a!xdx"=2a!x (21) x x!adx"=23a(x!a)3/2+25(x!a)5/2 (22) ax+bdx!=2b3a+2x3"#$%&'b+ax (23) (ax+b)3/2dx!=b+ax2b25a+4bx5+2ax25"#$%&' (24) xx a!dx=23x 2a()x a (25) xa!xdx=!xa!x"!atan!1xa!xx!a#$%&'( (26) xx+adx=xx+a!"alnx+x+a#$%& (27) x ax+bdx!="4b215a2+2bx15a+2x25#$%&'(b+ax (28) xax+bdx!=b x4a+x3/22"#$%&'b+ax!!!!!!!!!!!!!!!!!!!!! !!!!(b2ln 2ax+2b+ax()4a3/2 (29) x3/2ax+bdx!="b2x8a2+bx3/212a+x5/23#$%&'( b+ax"b3ln 2ax+2b+ax()8a5/2 (30) x2 a2!))))))
3 Dx=12x x2 a2 12a2lnx+x2 a2() (31) a2!x2"dx=12x a2!x2!12a2tan!1x a2!x2x2!a2#$%&'( (32) x x2 a2!=13(x2 a2)3/2 (33) 1x2 a2dx=lnx+x2 a2()! 2005 BE Shapiro Page 2 This document may not be reproduced, posted or published without permission. The copyright holder makes no representation about the accuracy, correctness, or suitability of this material for any purpose. (34) 1a2!x2"=sin!1xa (35) xx2 a2=x2 a2! (36) xa2!x2"dx=!a2!x2 (37) x2x2 a2!dx=12x x2 a2!12lnx+x2 a2() (38) x2a2!x2"dx=!12x a!x2!12a2tan!1x a2!x2x2!a2#$%&'( (39) ax2+bx+c!!dx=b4a+x2"#$%&'ax2+bx+c!!!!!!! !!!!!!!+4ac(b28a3/2ln2ax+ba+2ax2+bc+c"#$ %&' (40) x ax2+bx+c!dx!=!!!!!!!!!!!!!!!x33+bx12a+8a c"3b224a2#$%&'(ax2+bx+c!))))
4 !!!!!!!!!!!!!"b(4ac"b2)16a5/2ln2ax+ba+2a x2+bc+c#$%&'( (41) 1ax2+bx+c!dx=1aln2ax+ba+2ax2+bx+c"#$%&' (42) xax2+bx+c!dx=1aax2+bx+c!!!!!"b2a3/2ln2ax +ba+2ax2+bx+c#$%&'( LOGARITHMS (43) lnx!dx=xlnx"x (44) ln(ax)xdx!=12ln(ax)()2 (45) ln(ax+b)!dx=ax+baln(ax+b)"x (46) ln(a2x2 b2!)dx=xln(a2x2 b2)+2batan"1axb#$%&'("2x (47) ln(a2!b2x2")dx=xln(a2!b2x2)+2abtan!1bxa# $%&'(!2x (48) ln(ax2+bx+c)dx!=1a4ac"b2tan"12ax+b4ac"b2 #$%&'(!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! !"2x+b2a+x#$%&'(lnax2+bx+c() (49) xln(ax+b)dx!=b2ax"14x2+12x2"b2a2#$%&'(ln (ax+b) (50) xln(a2!b2x2)dx"=!12x2+12x2!a2b2#$%&'(ln( a2!bx2) EXPONENTIALS (51) eaxdx!=1aeax (52) xeaxdx!=1axeax+i"2a3/2erfi ax() where erf(x)=2!))))))))
5 E"t2dt0x# (53) xexdx!=(x"1)ex (54) xeaxdx!=xa"1a2#$%&'(eax (55) x2exdx!=ex(x2"2x+2) (56) x2eaxdx!=eaxx2a"2xa2+2a3#$%&'( (57) x3exdx!=ex(x3"3x2+6x"6) (58) xneaxdx!="1( )n1a#[1+n,"ax]where !(a,x)=ta"1e"tdtx#$ (59) eax2dx!="i#2aerfixa() TRIGONOMETRIC FUNCTIONS (60) sinxdx!="cosx (61) sin2xdx!=x2"14sin2x (62) sin3xdx!="34cosx+112cos3x (63) cosxdx!=sinx (64) cos2xdx!=x2+14sin2x (65) cos3xdx!=34sinx+112sin3x (66) sinxcosxdx!="12cos2x 2005 BE Shapiro Page 3 This document may not be reproduced, posted or published without permission. The copyright holder makes no representation about the accuracy, correctness, or suitability of this material for any purpose.))
6 (67) sin2xcosxdx!=14sinx"112sin3x (68) sinxcos2xdx!="14cosx"112cos3x (69) sin2xcos2xdx!=x8"132sin4x (70) tanxdx!="lncosx (71) tan2xdx!="x+tanx (72) tan3xdx!=ln[cosx]+12sec2x (73) secxdx!=ln|secx+tanx| (74) sec2xdx!=tanx (75) sec3xdx!=12secxtanx+12ln | secxtanx| (76) secxtanxdx!=secx (77) sec2xtanxdx!=12sec2x (78) secnxtanxdx!=1nsecnx, n!0 (79) cscxdx!=ln|cscx"cotx| (80) csc2xdx=!"cotx (81) csc3xdx=!"12cotxcscx+12ln | cscx"cotx| (82) cscnxcotxdx!="1ncscnx, n!0 (83) secxcscxdx!=lntanx TRIGONOMETRIC FUNCTIONS WITH xn (84) xcosxdx!=cosx+xsinx (85) xcos(ax)dx!=1a2cosax+1axsinax (86) x2cosxdx!=2xcosx+(x2"2)sinx (87) x2cosaxdx!=2a2xcosax+a2x2"2a3sinax (88) xncosxdx!
7 =!!!!!!!!!"12i( )1+n#(1+n,"ix)+"1( )n#(1+n,ix)$%&' (89) xncosaxdx!=!!!!!!!!!!12(ia)1"n("1)n#(1+n ,"iax)"#(1+n,iax)$%&' (90) xsinxdx!="xcosx+sinx (91) xsin(ax)dx!="xacosax+1a2sinax (92) x2sinxdx!=(2"x2)cosx+2xsinx (93) x3sinaxdx!=2"a2x2a3cosax+2a3xsinax (94) xnsinxdx!="12(i)n#(n+1,"ix)"("1)n#(n+1," ix)$%&' TRIGONOMETRIC FUNCTIONS WITH eax (95) exsinxdx!=12exsinx"cosx[] (96) ebxsin(ax)dx!=1b2+a2ebxbsinax"acosax[] (97) excosxdx!=12exsinx+cosx[] (98) ebxcos(ax)dx!=1b2+a2ebxasinax+bcosax[] TRIGONOMETRIC FUNCTIONS WITH xnAND eax (99) xexsinxdx!=12excosx"xcosx+xsinx[] (100) xexcosxdx!=12exxcosx"sinx+xsinx[] HYPERBOLIC FUNCTIONS (101) coshxdx!=sinhx (102) eaxcoshbxdx!
8 =eaxa2"b2acoshbx"bsinhbx[] (103) sinhxdx!=coshx (104) eaxsinhbxdx!=eaxa2"b2"bcoshbx+asinhbx[] (105) extanhxdx!=ex"2tan"1(ex) (106) tanhaxdx!=1alncoshax (107) cosaxcoshbxdx!=!!!!!!!!!!1a2+b2asinaxcos hbx+bcosaxsinhbx[] 2005 BE Shapiro Page 4 This document may not be reproduced, posted or published without permission. The copyright holder makes no representation about the accuracy, correctness, or suitability of this material for any purpose. (108) cosaxsinhbxdx!=!!!!!!!!!!1a2+b2bcosaxcos hbx+asinaxsinhbx[] (109) sinaxcoshbxdx!=!!!!!!!!!!1a2+b2"acosaxco shbx+bsinaxsinhbx[] (110) sinaxsinhbxdx!=!!!!!!!!!!1a2+b2bcoshbxsi nax"acosaxsinhbx[] (111) sinhaxcoshaxdx!
9 =14a"2ax+sinh(2ax)[] (112) sinhaxcoshbxdx!=!!!!!!!!!!1b2"a2bcoshbxs inhax"acoshaxsinhbx[]
