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©2005 BE Sh apiro 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) (2) (3) (4) RATIONAL FUNCTIONS (5) (6) (7) , (8) (9) (10) (11) (12) (13) (15) , (16) (17) INTEGRALS WITH ROOTS (19) (20) (21) (22) (23) (24) (25) (26) (27) (28) (29) (30) (31) (32) (33) xndx!=1n+1 xn+1 1x dx!=lnx dxa2+x2 !=1a tan"1(x/a) xdxa2+x2 !=12 ln(a2+x2) x2dxa2+x2 !=x"atan"1(x/a) x3dxa2+x2 !=12 x2"1 a2ln(a2+x2) (ax2+bx+c)!1dx"=24ac!b2 tan!12ax+b4ac!b2 #$%&'( 1(x+a)(x+b) dx=!1b"a ln(a+x)"ln(b+x)[] a!b x(x+a)2 dx=!aa+x +ln(a+x) xax2+bx+c dx!=ln(ax2+bx+c)2a !!!!!"ba4ac"b2 tan"12ax+b4ac"b2 #$%&'( x!a dx"=23 (x!a)3/2 udv!=uv"vdu! 1x±a dx!=2x±a 1a!x dx"=2a!x xx!a dx"=23 a(x!a)3/2+25 (x!a)5/2 ax+b dx!=2b3a +2x3 "#$%&'b+ax (ax+b)3/2dx!=b+ax 2b25a +4bx5 +2ax25 "#$%&' xx±a !dx=23 x±2a()x±a xa!x dx=!x a!x "!atan!1x a!x x!a #$%&'( xx+a dx=x x+a !"alnx +x+a #$% xax+b dx!="4b215a2 +2bx15a +2x25 #$%&'(b+ax x ax+b dx!=bx 4a +x3/22 "#$%&'b+ax !!!!!!!!!!!!!!!!!!!!!!!!!(b2ln2a x +2b+ax ()4a3/2 u(x)!v(x)dx"=u(x)v(x)#v(x)!u(x)dx" x3/2ax+b dx!="b2x 8a2 +bx3/212a +x5/23 #$%&'(b+ax "b3ln2a x +2b+ax ()8a5/2 x2±a2 !dx=12 xx2±a2 ±1 a2lnx+x2±a2 () a2!x2 "dx=12 xa2!x2 !1 a2tan!1xa2!x2 x2!a2 #$%&'( xx2±a2 !=13 (x2±a2)3/2 1x2±a2 dx=lnx+x2±a2 ()! 1ax+b dx!=1a ln(ax+b) 1(x+a)2 dx!="1x+a (x+a)ndx!=(x+a)na1+n +x1+n "#$%&' n!"1 x(x+a)ndx!=(x+a)1+n(nx+x"a)(n+2)(n+1) dx1+x2 !=tan"1x ©2005 BE Sh apiro Page 2 This document may not be reproduced, posted or published without permission. The copyright holder makes no representation about the accuracy, correctness, or suitability of this material for any purpose. (34) (35) (36) (37) (38) (39) (40) (41) (42) LOGARITHMS (43) (44) (45) (46) (47) (48) (49) (50) EXPONENTIALS (51) (52) where (53) (54) (55) (56) (57) (58) where (59) TRIGONOMETRIC FUNCTI ONS (60) (61) (62) (63) (64) (65) (66) 1a2!x2 "=sin!1xa xx2±a2 =x2±a2 ! ln(ax)x dx!=12 ln(ax)()2 ln(ax+b)!dx=ax+ba ln(ax+b)"x ln(a2x2±b2!)dx=xln(a2x2±b2)+2ba tan"1axb #$%&'("2x ln(a2!b2x2")dx=xln(a2!b2x2)+2ab tan!1bxa #$%&'(!2x ln(ax2+bx+c)dx!=1a 4ac"b2 tan"12ax+b4ac"b2 #$%&'(!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!"2x+b2a +x#$%&'(lnax2+bx+c() xln(ax+b)dx!=b2a x"14 x2+1 x2"b2a2 #$%&'(ln(ax+b) xln(a2!b2x2)dx"=!12 x2+1 x2!a2b2 #$%&'(ln(a2!bx2) eaxdx!=1a eax x eaxdx!=1a x eax+i" 2a3/2 erfiax () erf(x)=2! e"t2dt0x# xa2!x2 "dx=!a2!x2 xexdx!=(x"1)ex xeaxdx!=xa "1a2 #$%&'(eax x2exdx!=ex(x2"2x+2) x2eaxdx!=eaxx2a "2xa2 +2a3 #$%&'( x3exdx!=ex(x3"3x2+6x"6) xneaxdx!="1()n1a #[1+n,"ax] !(a,x)=ta"1e"tdtx#$ eax2dx!="i# 2a erfixa () sinxdx!="cosx sin2xdx!=x2 "14 sin2x x2x2±a2 !dx=12 xx2±a2 !1 lnx+x2±a2 () sin3xdx!="34 cosx+112 cos3x cosxdx!=sinx cos2xdx!=x2 +14 sin2x cos3xdx!=34 sinx+112 sin3x sinxcosxdx!="12 cos2x x2a2!x2 "dx=!12 xa!x2 !1 a2tan!1xa2!x2 x2!a2 #$%&'( ax2+bx+c !!dx=b4a +x2 "#$%&'ax2+bx+c !!!!!!!!!!!!!!+4ac(b28a3/2 ln2ax+ba +2ax2+bc+c "#$%&' xax2+bx+c !dx!=!!!!!!!!!!!!!!!x33 +bx12a +8ac"3b224a2 #$%&'(ax2+bx+c !!!!!!!!!!!!!!"b(4ac"b2)16a5/2 ln2ax+ba +2ax2+bc+c #$%&'( 1ax2+bx+c !dx=1a ln2ax+ba +2ax2+bx+c "#$%&' xax2+bx+c !dx=1a ax2+bx+c !!!!!"b2a3/2 ln2ax+ba +2ax2+bx+c #$%&'( lnx!dx=xlnx"x ©2005 BE Sh apiro Page 3 This document may not be reproduced, posted or published without permission. The copyright holder makes no representation about the accuracy, correctness, or suitability of this material for any purpose. (67) (68) (69) (70) (71) (72) (73) (74) (75) (76) (77) (78) , (79) (80) (81) (82) , (83) TRIGONOMETRIC FUNCTI ONS WITH (84) (85) (86) (87) (88) (89) (90) (91) (92) (93) (94) TRIGONOMETRIC FUNCTI ONS WITH (95) (96) (97) (98) TRIGONOMETRIC FUNCTI ONS WITH AND (99) (100) HYPERBOLIC FUNCTIONS (101) (102) (103) (104) (105) (106) (107) sin2xcosxdx!=14 sinx"112 sin3x sinxcos2xdx!="14 cosx"112 cos3x sec2xtanxdx!=12 sec2x secnxtanxdx!=1n secnx n!0 cscxdx!=ln|cscx"cotx| csc2xdx=!"cotx csc3xdx=!"12 cotxcscx+1 ln|cscx"cotx| cscnxcotxdx!="1n cscnx n!0 secxcscxdx!=lntanx xn sin2xcos2xdx!=x8 "132 sin4x xcosxdx!=cosx+xsinx xcos(ax)dx!=1a2 cosax+1a xsinax x2cosxdx!=2xcosx+(x2"2)sinx x2cosaxdx!=2a2 xcosax+a2x2"2a3 sinax xncosxdx!=!!!!!!!!!"12 i()1+n#(1+n,"ix)+"1()n#(1+n,ix)$%& xncosaxdx!=!!!!!!!!!!12 (ia)1"n("1)n#(1+n,"iax)"#(1+n,iax)$%& xsinxdx!="xcosx+sinx xsin(ax)dx!="xa cosax+1a2 sinax x2sinxdx!=(2"x2)cosx+2xsinx x sinaxdx ! =2"a2x2a3 cos sin tanxdx!="lncosx xnsinxdx!="12 (i)n#(n+1,"ix)"("1)n#(n+1,"ix)$%& eax exsinxdx!=12 exsinx"cosx[] ebxsin(ax)dx!=1b2+a2 ebxbsinax"acosax[] excosxdx!=12 exsinx+cosx[] ebxcos(ax)dx!=1b2+a2 ebxasinax+bcosax[] xn eax xexsinxdx!=12 excosx"xcosx+xsinx[] xexcosxdx!=12 exxcosx"sinx+xsinx[] tan2xdx!="x+tanx coshxdx!=sinhx eaxcoshbxdx!=eaxa2"b2 acoshbx"bsinhbx[] sinhxdx!=coshx eaxsinhbxdx!=eaxa2"b2 "bcoshbx+asinhbx[] extanhxdx!=ex"2tan"1(ex) tanhaxdx!=1a lncoshax cosaxcoshbxdx!=!!!!!!!!!!1a2+b2 asinaxcoshbx+bcosaxsinhbx[] tan3xdx!=ln[cosx]+12 sec2x secxdx!=ln|secx+tanx| sec2xdx!=tanx sec3xdx!=12 secxtanx+1 ln|secxtanx| secxtanxdx!=secx ©2005 BE Sh apiro 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) (109) (110) (111) (112) cosaxsinhbxdx!=!!!!!!!!!!1a2+b2 bcosaxcoshbx+asinaxsinhbx[] sinaxcoshbxdx!=!!!!!!!!!!1a2+b2 "acosaxcoshbx+bsinaxsinhbx[] sinaxsinhbxdx!=!!!!!!!!!!1a2+b2 bcoshbxsinax"acosaxsinhbx[] sinhaxcoshaxdx!=14a "2ax+sinh(2ax)[] sinhaxcoshbxdx!=!!!!!!!!!!1b2"a2 bcoshbxsinhax"acoshaxsinhbx[]