Table of Basic Integrals

Transcription

1 Tble of Bsic Integrls Bsic Forms () n d = n + n+, n () d = ln (3) udv = uv vdu (4) + b d = ln + b Integrls of Rtionl Functions (5) ( + ) d = + (6) ( + ) n d = ( + )n+, n n + (7) ( + ) n d = ( + )n+ ((n + ) ) (n + )(n + ) (8) + d = tn (9) + d = tn

2 (0) () + d = ln + + d = tn () 3 + d = ln + (3) + b + c d = + b 4c b tn 4c b (4) ( + )( + b) d = b ln + b +, b (5) ( + ) d = + ln + + (6) + b + c d = ln +b+c b 4c b tn + b 4c b Integrls with Roots (7) d = ( )3/ 3 (8) ± d = ± (9) d =

3 (0) d = ( 3 )3/ + ( 5 )5/, or ( 3 )3/ 4 ( 5 )5/, or ( + 3)( )3/ 5 () + b d = ( b ) + b () (3) ( + b) 3/ d = ( + b)5/ 5 ± d = 3 ( ) ± (4) (5) ( ) d = ( ) tn + d = ( + ) ln [ + + ] (6) + b d = 5 ( b + b + 3 ) + b (7) ( [ + b) d = ( + b) ( + b) b ln + ] ( + b) 4 3/ (8) [ 3 b ( + b) d = b 8 + ] 3 ( + b)+ b3 3 8 ln + ( + b) 5/ (9) ± d = ± ± ln + ± 3

4 (30) d = + tn (3) ± d = 3 ( ± ) 3/ (3) ± d = ln + ± (33) d = sin (34) ± d = ± (35) d = (36) ± d = ± ln + ± (37) + b + c d = b + 4c b + b + c+ ln + b + ( b + c) 3/ (38) + b + c d = ( 48 + b + c ( 3b + b + 8(c + ) ) 5/ +3(b 3 4bc) ln b + + ) + b + c 4

5 (39) + b + c d = ln + b + ( + b + c) (40) + b + c d = + b + c b ln + b + ( + b + c) 3/ (4) d ( + ) = 3/ + Integrls with Logrithms (4) ln d = ln (43) ln d = ln 4 (44) ln d = 3 3 ln 3 9 (45) ( ) ln n ln d = n+ n +, n (n + ) (46) ln d = (ln ) (47) ln d = ln 5

6 (48) ln( + b) d = ( + b ) ln( + b), 0 (49) ln( + ) d = ln( + ) + tn (50) ln( ) d = ln( ) + ln + (5) ln ( + b + c ) d = ( ) 4c b tn + b b + 4c b + ln ( + b + c ) (5) ln( + b) d = b 4 + ( b ) ln( + b) (53) ln ( b ) d = + ( b ) ln ( b ) (54) (ln ) d = ln + (ln ) (55) (ln ) 3 d = 6 + (ln ) 3 3(ln ) + 6 ln (56) (ln ) d = 4 + (ln ) ln (57) (ln ) d = (ln ) 9 3 ln 6

7 Integrls with Eponentils (58) e d = e (59) e d = e + i π erf ( i ), where erf() = 3/ π 0 e t dt (60) e d = ( )e (6) (6) e d = ( ) e e d = ( + ) e (63) (64) ( e d = + 3 ) e 3 e d = ( ) e (65) n e d = n e n n e d (66) n e d = ( )n Γ[ + n, ], where Γ(, ) = n+ t e t dt (67) e d = i π erf ( i ) 7

8 (68) e d = π erf ( ) (69) e d = e (70) e d = 4 π 3 erf( ) e Integrls with Trigonometric Functions (7) sin d = cos (7) sin d = sin 4 (73) sin 3 3 cos cos 3 d = + 4 (74) sin n d = [ cos F, n, 3 ], cos (75) cos d = sin (76) cos d = + sin 4 (77) cos 3 d = 3 sin 4 + sin 3 8

9 (78) (79) cos p d = ( + p) cos+p F [ + p,, 3 + p ], cos cos sin d = sin + c = cos + c = 4 cos + c 3 (80) cos sin b d = cos[( b)] ( b) cos[( + b)], b ( + b) (8) sin sin[( b)] sin b sin[( + b)] cos b d = + 4( b) b 4( + b) (8) sin cos d = 3 sin3 (83) cos sin b d = cos[( b)] 4( b) cos b b cos[( + b)] 4( + b) (84) cos sin d = 3 cos3 (85) sin cos bd = sin sin[( b)] sin b sin[( + b)] ( b) 8b 6( + b) (86) sin cos d = 8 sin 4 3 (87) tn d = ln cos 9

10 (88) tn d = + tn (89) ( tn n d = tnn+ n + ( + n) F,, n + 3 ), tn (90) tn 3 d = ln cos + sec (9) ( sec d = ln sec + tn = tnh tn ) (9) sec d = tn (93) sec 3 d = sec tn + ln sec + tn (94) sec tn d = sec (95) sec tn d = sec (96) sec n tn d = n secn, n 0 (97) csc d = ln tn = ln csc cot + C 0

11 (98) csc d = cot (99) csc 3 d = cot csc + ln csc cot (00) csc n cot d = n cscn, n 0 (0) sec csc d = ln tn Products of Trigonometric Functions nd Monomils (0) cos d = cos + sin (03) cos d = cos + sin (04) cos d = cos + ( ) sin (05) cos d = cos + sin 3 (06) n cos d = (i)n+ [Γ(n +, i) + ( ) n Γ(n +, i)]

12 (07) n cos d = (i) n [( ) n Γ(n +, i) Γ(n +, i)] (08) sin d = cos + sin (09) cos sin sin d = + (0) sin d = ( ) cos + sin () sin d = cos + 3 sin () n sin d = (i)n [Γ(n +, i) ( ) n Γ(n +, i)] (3) (4) cos d = cos + sin 4 sin d = 4 8 cos sin 4 (5) tn d = + ln cos + tn (6) sec d = ln cos + tn

13 Products of Trigonometric Functions nd Eponentils (7) e sin d = e (sin cos ) (8) e b sin d = + b eb (b sin cos ) (9) e cos d = e (sin + cos ) (0) () e b cos d = + b eb ( sin + b cos ) e sin d = e (cos cos + sin ) () e cos d = e ( cos sin + sin ) Integrls of Hyperbolic Functions (3) cosh d = sinh (4) e [ cosh b b sinh b] e cosh b d = b e 4 + b = b (5) sinh d = cosh 3

14 (6) (7) e [ b cosh b + sinh b] e sinh b d = b e 4 tnh d = ln cosh b = b (8) (9) (30) (3) (3) (33) (34) e (+b) [ ( + b) F + b,, + ] b, eb e tnh b d = [ e F, b, + ] b, eb cos cosh b d = cos sinh b d = sin cosh b d = sin sinh b d = e tn [e ] b = b [ sin cosh b + b cos sinh b] + b [b cos cosh b + sin sinh b] + b [ cos cosh b + b sin sinh b] + b [b cosh b sin cos sinh b] + b sinh cosh d = [ + sinh ] 4 sinh cosh b d = [b cosh b sinh cosh sinh b] b c 04. From lst revised June 4, 04. This mteril is provided s is without wrrnty or representtion bout the ccurcy, correctness or suitbility of this mteril for ny purpose. This work is licensed under the Cretive Commons Attribution-Noncommercil-Shre Alike 3.0 United Sttes License. To view copy of this license, visit or send letter to Cretive Commons, 7 Second Street, Suite 300, Sn Frncisco, Cliforni, 9405, USA. 4