Integral table for electomagnetic

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Published on February 25, 2014

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tabel integral

Table of Integrals BASIC FORMS INTEGRALS WITH ROOTS 1 (1) ! x dx = n + 1 x (2) ! x dx = ln x (3) ! udv = uv " ! vdu (4) " u(x)v!(x)dx = u(x)v(x) # " v(x)u !(x)dx n " x ! adx = (19) ! 1 dx = 2 x ± a x±a (20) " 1 dx = 2 a ! x a! x (21) "x (22) ! (23) 1 RATIONAL FUNCTIONS (5) ! (ax + b) 1 1 ! ax + b dx = a ln(ax + b) 1 2 (x ! a)3/2 3 (18) n+1 "1 x+a x ! adx = 2 2 a(x ! a)3/2 + (x ! a)5/2 3 5 " 2b 2x % b + ax ax + bdx = \$ + # 3a 3 ' & 3/2 " 2b 2 4bx 2ax 2 % dx = b + ax \$ + + 5 ' 5 # 5a & (6) ! (x + a) (24) (7) x % " a ! (x + a) dx = (x + a) \$ 1+n + 1+ n ' , n ! "1 # & ! 2 x dx = ( x ± 2a ) x ± a 3 x±a (25) (8) (x + a)1+n (nx + x " a) ! x(x + a) dx = (n + 2)(n + 1) " # x a! x& x dx = ! x a ! x ! a tan !1 % ( a! x \$ x!a ' (9) dx "1 ! 1+ x 2 = tan x (26) ! x dx = x x + a " a ln # x + x + a % \$ & x+a (10) 1 "1 dx ! a 2 + x 2 = a tan (x / a) (27) !x (11) !a (12) x 2 dx "1 ! a 2 + x 2 = x " a tan (x / a) 2 dx = n n n 1 xdx = ln(a 2 + x 2 ) + x2 2 ! 2 (28) (13) (14) " (ax + bx + c)!1 dx = !x # 2ax + b & tan !1 % \$ 4ac ! b 2 ( ' 4ac ! b 2 (15) (16) ! (x + a) (17) ! ax 2 2 dx = ln(ax 2 + bx + c) x dx = + bx + c 2a ©2005 BE Shapiro 4a ) 3/2 (29) " ( b 3 ln 2 a x + 2 b + ax 8a ) 5/2 ( ) # 2ax + b & tan "1 % \$ 4ac " b 2 ( ' a 4ac " b (30) ! x 2 ± a 2 dx = 1 1 x x 2 ± a 2 ± a 2 ln x + x 2 ± a 2 2 2 (31) a + ln(a + x) a+ x !!!!!" ( b 2 ln 2 a x + 2 b + ax # b 2 x bx 3/2 x 5/2 & b + ax ax + bdx = % " + + 2 12a 3 ( \$ 8a ' 3/2 2 1 1 ! (x + a)(x + b) dx = b " a [ ln(a + x) " ln(b + x)] , a ! b x " b x x 3/2 % b + ax x ax + bdx = \$ + 2 ' # 4a & !!!!!!!!!!!!!!!!!!!!!!!!!( 1 2 1 2 x 3 dx 2 2 ! a 2 + x 2 = 2 x " 2 a ln(a + x ) 2 # 4b 2 2bx 2x 2 & ax + bdx = % " + + b + ax 5 ( \$ 15a 2 15a ' " a 2 ! x 2 dx = # x a2 ! x2 & 1 1 x a 2 ! x 2 ! a 2 tan !1 % 2 ( 2 2 2 \$ x !a ' (32) !x (33) ! b 2 1 x 2 ± a 2 = (x 2 ± a 2 )3/2 3 1 x ±a 2 2 ( dx = ln x + x 2 ± a 2 ) 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.

1 (34) " a !x (35) ! x2 ± a2 (36) " a2 ! x2 (37) x x ±a (40) dx = 2 ( 1 1 x x 2 ± a 2 ! ln x + x 2 ± a 2 2 2 ) (51) " b x% ax 2 + bx + c ax 2 + bx + c !dx = \$ + # 4a 2 ' & !e (52) ! 1 1# a2 & ! b 2 x 2 )dx = ! x 2 + % x 2 ! 2 ( ln(a 2 ! bx 2 ) b ' 2 2\$ ax dx = 1 ax e a 1 i " xeax + 3/2 erf i ax 2a a ( xeax dx = 2 ! # x 0 ! xe (55) ! x e dx = e (x (56) b(4ac " b ) # 2ax + b & ln % + 2 ax 2 + bc + c ( \$ ' 16a 5/2 a 1 " 2ax + b % dx = ln + 2 ax 2 + bx + c ' a \$ a # & ax 2 + bx + c 1 1 x dx = ax 2 + bx + c a ax 2 + bx + c b # 2ax + b & !!!!!" 3/2 ln % + 2 ax 2 + bx + c ( 2a a \$ ' where 2 (54) # x 3 bx 8ac " 3b 2 & + ax 2 + bx + c !!!!!!!!!!!!!!! % + 24a 2 ( \$ 3 12a ' ) e"t dt ! xe dx = (x " 1)e # x 2 2x 2 & x 2 eax dx = eax % " 2 + 3 ( ! a ' \$ a a (57) ! x e dx = e (x (58) !x e ax 2 + bx + c !dx = !!!!!!!!!!!!!!" ! (42) 1# b2 & + % x 2 " 2 ( ln(ax + b) a ' 2\$ (53) 4ac ( b 2 " 2ax + b % !!!!!!!!!!!!!!+ ln \$ + 2 ax 2 + bc + c ' # & 8a 3/2 a ! 2 erf (x) = 2 (41) " x ln(a 2 EXPONENTIALS # x a2 ! x2 & 1 1 dx = ! x a ! x 2 ! a 2 tan !1 % 2 ( 2 2 2 a2 ! x2 \$ x !a ' !x 1 ! x ln(ax + b)dx = 2a x " 4 x (50) x2 ! b (49) dx = ! a 2 ! x 2 2 2 x a = x 2 ± a2 x " (39) = sin !1 2 x ! (38) 2 x ax #x 1& dx = % " 2 ( eax \$a a ' 2 x x 3 x x n ax !e ax 2 2 3 dx = ( "1) !(a, x) = (59) x \$ # x dx = "i n " 2x + 2) " 3x 2 + 6x " 6) 1 #[1+ n, "ax] where a t a"1e"t dt # erf ix a 2 a ( ) LOGARITHMS (43) ! ln xdx = x ln x " x (44) ! (45) ! ln(ax + b)dx = (46) 2b "1 # ax & ! ln(a x ± b )dx = x ln(a x ± b ) + a tan % b ( " 2x \$ ' (47) 2a !1 # bx & " ln(a ! b x )dx = x ln(a ! b x ) + b tan % a ( ! 2x \$ ' (48) TRIGONOMETRIC FUNCTIONS (60) 2 2 2 ax + b ln(ax + b) " x a 2 2 2 2 2 2 2 2 2 + bx + c)dx = ©2005 BE Shapiro ! sin (62) ! sin (63) ! cos xdx = sin x (64) ! cos (65) ! cos (66) ! sin x cos xdx = " 2 cos 2 # 2ax + b & 1 4ac " b 2 tan "1 % a \$ 4ac " b 2 ( ' # b & !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!"2x + % + x ( ln ax 2 + bx + c \$ 2a ' ! ln(ax ! sin xdx = " cos x (61) 1 ln(ax) 2 dx = ( ln(ax)) 2 x ( ) 2 3 xdx = x 1 " sin 2x 2 4 3 1 xdx = " cos x + cos 3x 4 12 2 xdx = x 1 + sin 2x 2 4 3 xdx = 3 1 sin x + sin 3x 4 12 1 2 x 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.

(67) ! sin 2 x cos xdx = 1 1 sin x " sin 3x 4 12 (68) 2 2 ! sin x cos xdx = 1 !!!!!!!!!! (ia)1"n \$("1)n #(1+ n, "iax) " #(1+ n,iax) & % ' 2 (90) ! x sin xdx = "x cos x + sin x (91) ! x sin(ax)dx = " a cos ax + a (92) !x (93) 3 ! x sin axdx = !x 2 x 1 " sin 4 x 8 32 (70) ! tan xdx = " ln cos x (71) 2 ! tan xdx = "x + tan x 1 xdx = ln[cos x] + sec 2 x 2 cos axdx = n (89) 1 1 ! sin x cos xdx = " 4 cos x " 12 cos 3x (69) !x x 1 2 sin ax sin xdx = (2 " x 2 )cos x + 2x sin x 2 2 " a2 x2 2 cos ax + 3 x sin ax a3 a (72) ! tan (73) ! sec xdx = ln | sec x + tan x | (94) (74) ! sec TRIGONOMETRIC FUNCTIONS WITH e ax (75) 1 1 ! sec xdx = 2 sec x tan x + 2 ln | sec x tan x | 3 2 xdx = tan x (95) !e (96) !e 3 1 sin xdx = " (i)n \$ #(n + 1, "ix) " ("1)n #(n + 1, "ix) & % ' 2 n x sin xdx = 1 x e [ sin x " cos x ] 2 sin(ax)dx = bx 1 ebx [ b sin ax " a cos ax ] b + a2 (76) ! sec x tan xdx = sec x (77) ! sec (78) ! sec (79) ! csc xdx = ln | csc x " cot x | TRIGONOMETRIC FUNCTIONS WITH x n AND e ax (80) ! csc (99) ! xe (81) ! csc (100) ! xe (82) ! csc (83) ! sec x csc xdx = ln tan x 2 x tan xdx = 1 2 sec x 2 (97) !e n x tan xdx = 1 n sec x , n ! 0 n (98) !e 2 3 n xdx = " cot x 1 1 xdx = " cot x csc x + ln | csc x " cot x | 2 2 x 1 x e [ sin x + cos x ] 2 cos xdx = cos(ax)dx = bx 1 ebx [ a sin ax + b cos ax ] b2 + a2 x sin xdx = 1 x e [ cos x " x cos x + x sin x ] 2 x cos xdx = 1 x e [ x cos x " sin x + x sin x ] 2 1 x cot xdx = " csc n x , n ! 0 n TRIGONOMETRIC FUNCTIONS WITH x n 2 HYPERBOLIC FUNCTIONS (101) ! cosh xdx = sinh x (102) !e ax cosh bxdx = eax [ a cosh bx " b sinh bx ] a " b2 2 (84) ! x cos xdx = cos x + x sin x (103) ! sinh xdx = cosh x (85) 1 1 ! x cos(ax)dx = a 2 cos ax + a x sin ax (104) !e (86) !x (105) !e (87) !x (106) ! tanh axdx = a ln cosh ax (88) !x 2 2 n cos xdx = 2x cos x + (x 2 " 2)sin x cos axdx = 2 a2 x2 " 2 x cos ax + sin ax 2 a a3 cos xdx = !!!!!!!!!" 1 1+n \$ (i ) % #(1+ n, "ix) + ( "1)n #(1+ n,ix)& ' 2 ©2005 BE Shapiro (107) ax x sinh bxdx = eax [ "b cosh bx + a sinh bx ] a " b2 2 tanh xdx = e x " 2 tan "1 (e x ) 1 ! cos ax cosh bxdx = !!!!!!!!!! 1 [ a sin ax cosh bx + b cos ax sinh bx ] a + b2 2 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.

(108) ! cos ax sinh bxdx = !!!!!!!!!! (109) ! sin ax cosh bxdx = !!!!!!!!!! (110) (112) 1 [ "a cos ax cosh bx + b sin ax sinh bx ] a + b2 2 ! sin ax sinh bxdx = !!!!!!!!!! (111) 1 [b cos ax cosh bx + a sin ax sinh bx ] a + b2 2 1 [b cosh bx sin ax " a cos ax sinh bx ] a + b2 2 1 ! sinh ax cosh axdx = 4a [ "2ax + sinh(2ax)] ! sinh ax cosh bxdx = !!!!!!!!!! 1 [b cosh bx sinh ax " a cosh ax sinh bx ] b2 " a2 ©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.

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