advertisement

Integral table for electomagnetic

50 %
50 %
advertisement
Information about Integral table for electomagnetic
Education

Published on February 25, 2014

Author: fathascoutaq

Source: slideshare.net

Description

tabel integral
advertisement

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.

Add a comment

Related presentations

Related pages

Faraday's Electromagnetic Lab - Magnetism, Magnetic Field ...

Faraday’s Electromagnetic Lab: Dr. Wendy Adams: UG-Intro HS: HW Guided: Investigating Electromagnetism Applications: Elyse Zimmer: HS: Lab: Exploring ...
Read more

Computational electromagnetics - Wikipedia, the free ...

Computational electromagnetics, computational electrodynamics or electromagnetic modeling is the process of modeling the interaction of electromagnetic ...
Read more

Integral formalism for surface electromagnetic waves in ...

Integral formalism for surface electromagnetic waves in bianisotropic media View the table of contents for this issue, ...
Read more

A stable well-conditioned integral equation for ...

Finding a formulation for electromagnetic scattering of ... Electromagnetic integral ... Journal of Computational and Applied Mathematics table of ...
Read more

A Finite Element-Boundary Integral Formulation for ...

A hybrid technique is presented that combines the finite element and boundary integral methods for simulating electromagnetic scattering from ...
Read more

Integral Equation Methods for Electromagnetic and Elastic ...

Abstract Integral Equation Methods for Electromagnetic and Elastic Waves is an outgrowth of several years of work. There have been no recent books on ...
Read more

Volume Integral Equations for Electromagnetic Scattering ...

Table II shows estimates ... volume EFIE for electromagnetic scattering from dielectric objects,” IEEE ... integral equation for electromagnetic ...
Read more

Electromagnetic Radiation - ChemWiki: The Dynamic ...

Introduction. Electromagnetic radiation is a form of energy that is produced by oscillating electric and magnetic disturbance, or by the movement of ...
Read more