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Table of Integrals

Table of Integrals

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Table of Integrals

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  1. Table of Integrals∗ Basic Forms Integrals with Logarithms Z x√ax + bdx = 15a2(−2b2+ abx + 3a2x2)√ax + b (26) 2 Z Z Z 1 xndx = n + 1xn+1 (1) lnaxdx = xlnax − x (42) Z 1 xdx = ln|x| lnax x dx =1 Zp 2(lnax)2 (2) h (43) p 1 x(ax + b)dx = (2ax + b) ???a√x + ax(ax + b) ??? ?p p Z ax + bdx =1 Z 4a3/2 i p ? ? Z udv = uv − vdu (3) −b2ln x +b a(ax + b) (27) ln(ax + b) − x,a 6= 0 ln(ax + b)dx = (44) Z a 1 aln|ax + b| (4) Z ? Zp ln(x2+ a2) dx = xln(x2+ a2) + 2atan−1x b2 b 8a2x+x ???a√x + a− 2x (45) Integrals of Rational Functions x3(ax + b)dx = x3(ax + b) 12a− b3 8a5/2ln 3 ??? Z Z + a(ax + b) (28) 1 1 ln(x2− a2) dx = xln(x2− a2) + alnx + a (x + a)2dx = − (5) x − a− 2x (46) x + a Z (x + a)ndx =(x + a)n+1 Z ,n 6= −1 Zp (6) ???x + x2± a2??? p p p ln?ax2+ bx + c?dx =1 − 2x + 2ax + b √4ac − b2 n + 1 x2± a2dx =1 x2± a2±1 4ac − b2tan−1 ln?ax2+ bx + c? 2a2ln 2x a Z ?b ? (29) x(x + a)ndx =(x + a)n+1((n + 1)x − a) (7) 2a+ x (47) (n + 1)(n + 2) Z Zp p 1 Z a2− x2dx =1 a2− x2+1 x 1 + x2dx = tan−1x (8) 2a2tan−1 √a2− x2 2x xln(ax + b)dx =bx 2a−1 +1 2 4x2 Z (30) ? ? a2+ x2dx =1 1 atan−1x x2−b2 (9) a ln(ax + b) (48) a2 Z Z Z p ?x2± a2?3/2 ???x + x2± a2dx =1 a2+ x2dx =1 x x (31) 2ln|a2+ x2| (10) 3 Z xln?a2− b2x2?dx = −1 2x2+ x2−a2 Z x2 x2± a2??? a2+ x2dx = x − atan−1x p 1 (11) ? ? √x2± a2dx = ln Z Z Z (32) a ln?a2− b2x2? 1 2 (49) Z b2 x3 a2+ x2dx =1 2x2−1 2a2ln|a2+ x2| (12) √a2− x2dx = sin−1x 1 (33) Integrals with Exponentials a Z 1 2 2ax + b √4ac − b2 √4ac − b2tan−1 ax2+ bx + cdx = (13) p p Z x eaxdx =1 √x2± a2dx = x2± a2 (34) aeax (50) Z 1 b − alna + x a a + x+ ln|a + x| 1 √xeax+i√π Z√xeaxdx =1 b + x, a 6= b (x + a)(x + b)dx = Z (14) x 2a3/2erf?i√ax?, 2 √π 0 √a2− x2dx = − a2− x2 (35) a Zx x (x + a)2dx = (15) e−t2dt where erf(x) = (51) Z ???x + x2± a2??? p p x2 √x2± a2dx =1 x2± a2∓1 2a2ln 2x Z Z x 1 2aln|ax2+ bx + c| b a√4ac − b2tan−1 Integrals with Roots xexdx = (x − 1)ex ?x x2exdx =?x2− 2x + 2?ex x2eaxdx = a ax2+ bx + cdx = (52) (36) ? Z 2ax + b √4ac − b2 − (16) a−1 xeaxdx = eax (53) Zp a2 p a(ax2+ bx+c) ax2+ bx + cdx =b + 2ax ax2+ bx + c Z 4a (54) ???2ax + b + 2 ??? p +4ac − b2 8a3/2 Z√x − adx =2 Z Z ln (37) ?x2 ? Z Z xneaxdx =xneax 3(x − a)3/2 (17) −2x a2+2 eax (55) a3 √x ± adx = 2√x ± a 1 √a − xdx = −2√a − x 1 Z x3exdx =?x3− 3x2+ 6x − 6?ex (18) ? p p p 2√a 1 (56) ax2+ bx + c = ax2+ bx + c x ×?−3b2+ 2abx + 8a(c + ax2)? +3(b3− 4abc)ln 48a5/2 Z Z (19) −n xn−1eaxdx (57) ???b + 2ax + 2√a ??? ? a a ax2+ bx + c (38) Z x√x − adx =2 3a(x − a)3/2+2 ?2b Z 5(x − a)5/2 ?√ax + b (20) xneaxdx =(−1)n an+1Γ[1 + n,−ax], Z∞ 2√aerf?ix√a? e−ax2dx = Z√ax + bdx = Z Z (58) Z ???2ax + b + 2 ??? 3a+2x p 1 1 ta−1e−tdt (21) where Γ(a,x) = √ax2+ bx + cdx = √aln a(ax2+ bx + c) 3 x eax2dx = −i√π Z Z (39) 2 (ax + b)3/2dx = 5a(ax + b)5/2 (22) (59) √π 2√aerf?x√a? 2ae−ax2 3(x ∓ 2a)√x ± a Z √x ± adx =2 x p (23) √ax2+ bx + cdx =1 b 2a3/2ln x (60) ax2+ bx + c a Z ???2ax + b + 2 dx (a2+ x2)3/2= ??? p px(a − x) Z r xe−ax2dx = −1 p a(ax2+ bx + c) − (40) x (61) x(a − x) − atan−1 a − xdx = − (24) x − a rπ Z Z a3erf(x√a) −x x x2e−ax2dx =1 Z r 2ae−ax2 p a2√a2+ x2 (41) (62) x(a + x) − aln?√x +√x + a? x 4 a + xdx = (25) ∗« 2014. From http://integral-table.com, last revised June 14, 2014. This material is provided as is without warranty or representation about the accuracy, correctness or suitability of the material for any purpose, and is licensed under the Creative Commons Attribution-Noncommercial-ShareAlike 3.0 United States License. To view a copy of this license, visit http://creativecommons.org/licenses/by-nc-sa/3.0/ or send a letter to Creative Commons, 171 Second Street, Suite 300, San Francisco, California, 94105, USA. 1

  2. Integrals with Trigonometric Functions Z excosxdx =1 Z 2ex(sinx + cosx) (106) sec3x dx =1 2secxtanx +1 2ln|secx + tanx| (84) Z sinaxdx = −1 acosax (63) Z Z sec2xtanxdx =1 1 ebxcosaxdx = a2+ b2ebx(asinax + bcosax) (107) Z secxtanxdx = secx (85) sin2axdx =x 2−sin2ax (64) Z 4a 2sec2x Z (86) xexsinxdx =1 Z 2ex(cosx − xcosx + xsinx) (108) Z sinnaxdx = secnxtanxdx =1 nsecnx,n 6= 0 (87) ?1 ? 2,1 − n −1 ,3 Z 2,cos2ax acosax 2F1 (65) xexcosxdx =1 2ex(xcosx − sinx + xsinx) 2 (109) Z ???tanx csc2axdx = −1 ??? = ln|cscx − cotx| + C acotax Z cscxdx = ln (88) 2 sin3axdx = −3cosax Z Z +cos3ax 12a (66) Integrals of Hyperbolic Functions 4a Z (89) cosaxdx =1 asinax (67) Z coshaxdx =1 asinhax (110) Z csc3xdx = −1 2cotxcscx +1 cos2axdx =x 2+sin2ax 2ln|cscx − cotx| (90) (68) 4a Z Z eaxcoshbxdx =   Z cscnxcotxdx = −1 Z ncscnx,n 6= 0 1 (91) cospaxdx = − a(1 + p)cos1+pax× ?1 + p eax    a2− b2[acoshbx − bsinhbx] e2ax 4a 2 Z a 6= b ? (111) ,1 2,3 + p secxcscxdx = ln|tanx| ,cos2ax (92) +x 2F1 (69) a = b 2 2 Z Products of Trigonometric Functions and Monomials cos3axdx =3sinax +sin3ax 12a sinhaxdx =1 (70) acoshax (112) 4a Z Z Z cosaxsinbxdx =cos[(a − b)x] −cos[(a + b)x] 2(a + b) ,a 6= b eaxsinhbxdx =   xcosxdx = cosx + xsinx (93) 2(a − b) (71) eax Z    1 a2cosax +x a2− b2[−bcoshbx + asinhbx] e2ax 4a 2 a 6= b xcosaxdx = asinax (94) (113) Z −x sin2axcosbxdx = −sin[(2a − b)x] +sinbx a = b Z x2cosxdx = 2xcosx +?x2− 2?sinx 4(2a − b) −sin[(2a + b)x] 4(2a + b) (95) Z (72) 2b eaxtanhbxdx =    Z +a2x2− 2 x2cosaxdx =2xcosax Z sinax (96) h 2b,−e2bxi e(a+2b)x (a + 2b)2F1 sin2xcosxdx =1 a2 a3   1 +a 2b,1,2 +a ha 3sin3x   (73)       2b,1,1E,−e2bxi −1 Z aeax2F1 (114) a 6= b a = b xncosxdx = −1 Z 2(i)n+1[Γ(n + 1,−ix) +(−1)nΓ(n + 1,ix)] cos2axsinbxdx =cos[(2a − b)x] −cosbx eax− 2tan−1[eax] a 4(2a − b) 2b (97) −cos[(2a + b)x] 4(2a + b) Z (74) tanhaxdx =1 alncoshax (115) Z Z xncosaxdx =1 2(ia)1−n[(−1)nΓ(n + 1,−iax) −Γ(n + 1,ixa)] cos2axsinaxdx = −1 3acos3ax (75) Z 1 cosaxcoshbxdx = a2+ b2[asinaxcoshbx +bcosaxsinhbx] (98) Z −sin[2(a − b)x] 16(a − b) sin2axcos2bxdx =x 4−sin2ax +sin2bx 8b (116) Z xsinaxdx = −xcosax 8a −sin[2(a + b)x] 16(a + b) xsinxdx = −xcosx + sinx (99) (76) Z Z 1 +sinax cosaxsinhbxdx = a2+ b2[bcosaxcoshbx+ asinaxsinhbx] (100) a2 a Z sin2axcos2axdx =x 8−sin4ax (117) (77) Z 32a x2sinxdx =?2 − x2?cosx + 2xsinx Z (101) tanaxdx = −1 alncosax (78) Z 1 a2+ b2[−acosaxcoshbx+ bsinaxsinhbx] sinaxcoshbxdx = Z Z x2sinaxdx =2 − a2x2 tan2axdx = −x +1 cosax +2xsinax atanax (79) (102) (118) a3 a2 Z Z tannaxdx =tann+1ax ?n + 1 Z xnsinxdx = −1 a(1 + n)× ,1,n + 3 2 1 2(i)n[Γ(n + 1,−ix) − (−1)nΓ(n + 1,−ix)] a2+ b2[bcoshbxsinax− acosaxsinhbx] sinaxsinhbxdx = ? (103) ,−tan2ax 2F1 (80) (119) 2 Products of Trigonometric Functions and Exponentials Z tan3axdx =1 1 2asec2ax Z alncosax + (81) 1 4a[−2ax + sinh2ax] sinhaxcoshaxdx = (120) Z Z secxdx = ln|secx + tanx| = 2tanh−1? Z ? exsinxdx =1 2ex(sinx − cosx) tanx (104) (82) Z 2 1 sinhaxcoshbxdx = b2− a2[bcoshbxsinhax −acoshaxsinhbx] Z sec2axdx =1 1 ebxsinaxdx = a2+ b2ebx(bsinax − acosax) atanax (83) (105) (121) 2

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