Processing math: 64%
 AI math handbook calculator - Fractional Calculus Computer Algebra System software Home | list | math | function | coding | graphics | example | help | 中文
+ + + =

Integral Table 2/2

<= prev page
  • Integrator

    Table of Basic Integrals

    Basic Forms

    xndx=1n+1xn+1,n1 (1)
    1xdx=ln|x| (2)
    udv=uvvdu (3)
    1ax+bdx=1aln|ax+b| (4)

    Integrals of Rational Functions

    1(x+a)2dx=1x+a (5)
    (x+a)ndx=(x+a)n+1n+1,n1 (6)
    x(x+a)ndx=(x+a)n+1((n+1)xa)(n+1)(n+2) (7)
    11+x2dx=tan1x (8)
    1a2+x2dx=1atan1xa (9)
    xa2+x2dx=12ln|a2+x2| (10)
    x2a2+x2dx=xatan1xa (11)
    x3a2+x2dx=12x212a2ln|a2+x2| (12)
    1ax2+bx+cdx=24acb2tan12ax+b4acb2 (13)
    1(x+a)(x+b)dx=1balna+xb+x, ab (14)
    x(x+a)2dx=aa+x+ln|a+x| (15)
    xax2+bx+cdx=12aln|ax2+bx+c|ba4acb2tan12ax+b4acb2 (16)

    Integrals with Roots

    xadx=23(xa)32 (17)
    1x±adx=2x±a (18)
    1axdx=2ax (19)
    xxadx={2a3(xa)32+25(xa)52, or23x(xa)32415(xa)52, or215(2a+3x)(xa)32 (20)
    ax+bdx=(2b3a+2x3)ax+b (21)
    (ax+b)32dx=25a(ax+b)52 (22)
    xx±adx=23(x2a)x±a (23)
    xaxdx=x(ax)atan1x(ax)xa (24)
    xa+xdx=x(a+x)aln[x+x+a] (25)
    xax+bdx=215a2(2b2+abx+3a2x2)ax+b (26)
    x(ax+b)dx=14a32[(2ax+b)ax(ax+b)b2ln|ax+a(ax+b)|] (27)
    x3(ax+b)dx=[b12ab28a2x+x3]x3(ax+b)+b38a52ln|ax+a(ax+b)| (28)
    x2±a2dx=12xx2±a2±12a2ln|x+x2±a2| (29)
    a2x2dx=12xa2x2+12a2tan1xa2x2 (30)
    xx2±a2dx=13(x2±a2)32 (31)
    1x2±a2dx=ln|x+x2±a2| (32)
    1a2x2dx=sin1xa (33)
    xx2±a2dx=x2±a2 (34)
    xa2x2dx=a2x2 (35)
    x2x2±a2dx=12xx2±a212a2ln|x+x2±a2| (36)
    ax2+bx+cdx=b+2ax4aax2+bx+c+4acb28a32ln|2ax+b+2a(ax2+bx+c)| (37)
    xax2+bx+cdx=148a52(2aax2+bx+c(3b2+2abx+8a(c+ax2))+3(b34abc)ln|b+2ax+2aax2+bx+c|) (38)
    1ax2+bx+cdx=1aln|2ax+b+2a(ax2+bx+c)| (39)
    xax2+bx+cdx=1aax2+bx+cb2a32ln|2ax+b+2a(ax2+bx+c)| (40)
    dx(a2+x2)32=xa2a2+x2 (41)

    Integrals with Logarithms

    lnaxdx=xlnaxx (42)
    xlnxdx=12x2lnxx24 (43)
    x2lnxdx=13x3lnxx39 (44)
    xnlnxdx=xn+1(lnxn+11(n+1)2),n1 (45)
    lnaxxdx=12(lnax)2 (46)
    lnxx2dx=1xlnxx (47)
    ln(ax+b)dx=(x+ba)ln(ax+b)x,a0 (48)
    ln(x2+a2)dx=xln(x2+a2)+2atan1xa2x (49)
    ln(x2a2)dx=xln(x2a2)+alnx+axa2x (50)
    ln(ax2+bx+c)dx=1a4acb2tan12ax+b4acb22x+(b2a+x)ln(ax2+bx+c) (51)
    xln(ax+b)dx=bx2a14x2+12(x2b2a2)ln(ax+b) (52)
    xln(a2b2x2)dx=12x2+12(x2a2b2)ln(a2b2x2) (53)
    (lnx)2dx=2x2xlnx+x(lnx)2 (54)
    (lnx)3dx=6x+x(lnx)33x(lnx)2+6xlnx (55)
    x(lnx)2dx=x24+12x2(lnx)212x2lnx (56)
    x2(lnx)2dx=2x327+13x3(lnx)229x3lnx (57)

    Integrals with Exponentials

    eaxdx=1aeax (58)
    xeaxdx=1axeax+iπ2a32erf(iax), where erf(x)=2πx0et2dt (59)
    xexdx=(x1)ex (60)
    xeaxdx=(xa1a2)eax (61)
    x2exdx=(x22x+2)ex (62)
    x2eaxdx=(x2a2xa2+2a3)eax (63)
    x3exdx=(x33x2+6x6)ex (64)
    xneaxdx=xneaxanaxn1eaxdx (65)
    xneaxdx=(1)nan+1Γ[1+n,ax], where Γ(a,x)=xta1etdt (66)
    eax2dx=iπ2aerf(ixa) (67)
    eax2dx=π2aerf(xa) (68)
    xeax2dx=12aeax2 (69)
    x2eax2dx=14πa3erf(xa)x2aeax2 (70)

    Integrals with Trigonometric Functions

    sinaxdx=1acosax (71)
    sin2axdx=x2sin2ax4a (72)
    sin3axdx=3cosax4a+cos3ax12a (73)
    sinnaxdx=1acosax2F1[12,1n2,32,cos2ax] (74)
    cosaxdx=1asinax (75)
    cos2axdx=x2+sin2ax4a (76)
    cos3axdx=3sinax4a+sin3ax12a (77)
    cospaxdx=1a(1+p)cos1+pax×2F1[1+p2,12,3+p2,cos2ax] (78)
    cosxsinxdx=12sin2x+c1=12cos2x+c2=14cos2x+c3 (79)
    cosaxsinbxdx=cos[(ab)x]2(ab)cos[(a+b)x]2(a+b),ab (80)
    sin2axcosbxdx=sin[(2ab)x]4(2ab)+sinbx2bsin[(2a+b)x]4(2a+b) (81)
    sin2xcosxdx=13sin3x (82)
    cos2axsinbxdx=cos[(2ab)x]4(2ab)cosbx2bcos[(2a+b)x]4(2a+b) (83)
    cos2axsinaxdx=13acos3ax (84)
    sin2axcos2bxdx=x4sin2ax8asin[2(ab)x]16(ab)+sin2bx8bsin[2(a+b)x]16(a+b) (85)
    sin2axcos2axdx=x8sin4ax32a (86)
    tan a x d x = 1 a ln cos a x (87)
    tan 2 a x d x = x + 1 a tan a x (88)
    tan n a x d x = tan n + 1 a x a ( 1 + n ) × 2 F 1 n + 1 2 , 1 , n + 3 2 , tan 2 a x (89)
    tan 3 a x d x = 1 a ln cos a x + 1 2 a sec 2 a x (90)
    sec x d x = ln | sec x + tan x | = 2 tanh 1 tan x 2 (91)
    sec 2 a x d x = 1 a tan a x (92)
    sec 3 x d x = 1 2 sec x tan x + 1 2 ln | sec x + tan x | (93)
    sec x tan x d x = sec x (94)
    sec 2 x tan x d x = 1 2 sec 2 x (95)
    sec n x tan x d x = 1 n sec n x , n 0 (96)
    csc x d x = ln tan x 2 = ln | csc x cot x | + C (97)
    csc 2 a x d x = 1 a cot a x (98)
    csc 3 x d x = 1 2 cot x csc x + 1 2 ln | csc x cot x | (99)
    csc n x cot x d x = 1 n csc n x , n 0 (100)
    sec x csc x d x = ln | tan x | (101)

    Products of Trigonometric Functions and Monomials

    x cos x d x = cos x + x sin x (102)
    x cos a x d x = 1 a 2 cos a x + x a sin a x (103)
    x 2 cos x d x = 2 x cos x + x 2 2 sin x (104)
    x 2 cos a x d x = 2 x cos a x a 2 + a 2 x 2 2 a 3 sin a x (105)
    x n cos x d x = 1 2 ( i ) n + 1 Γ ( n + 1 , i x ) + ( 1 ) n Γ ( n + 1 , i x ) (106)
    x n cos a x d x = 1 2 ( i a ) 1 n ( 1 ) n Γ ( n + 1 , i a x ) Γ ( n + 1 , i x a ) (107)
    x sin x d x = x cos x + sin x (108)
    x sin a x d x = x cos a x a + sin a x a 2 (109)
    x 2 sin x d x = 2 x 2 cos x + 2 x sin x (110)
    x 2 sin a x d x = 2 a 2 x 2 a 3 cos a x + 2 x sin a x a 2 (111)
    x n sin x d x = 1 2 ( i ) n Γ ( n + 1 , i x ) ( 1 ) n Γ ( n + 1 , i x ) (112)
    x cos 2 x d x = x 2 4 + 1 8 cos 2 x + 1 4 x sin 2 x (113)
    x sin 2 x d x = x 2 4 1 8 cos 2 x 1 4 x sin 2 x (114)
    x tan 2 x d x = x 2 2 + ln cos x + x tan x (115)
    x sec 2 x d x = ln cos x + x tan x (116)

    Products of Trigonometric Functions and Exponentials

    e x sin x d x = 1 2 e x ( sin x cos x ) (117)
    e b x sin a x d x = 1 a 2 + b 2 e b x ( b sin a x a cos a x ) (118)
    e x cos x d x = 1 2 e x ( sin x + cos x ) (119)
    e b x cos a x d x = 1 a 2 + b 2 e b x ( a sin a x + b cos a x ) (120)
    x e x sin x d x = 1 2 e x ( cos x x cos x + x sin x ) (121)
    x e x cos x d x = 1 2 e x ( x cos x sin x + x sin x ) (122)

    Integrals of Hyperbolic Functions

    cosh a x d x = 1 a sinh a x (123)
    e a x cosh b x d x = e a x a 2 b 2 [ a cosh b x b sinh b x ] a b e 2 a x 4 a + x 2 a = b (124)
    sinh a x d x = 1 a cosh a x (125)
    e a x sinh b x d x = e a x a 2 b 2 [ b cosh b x + a sinh b x ] a b e 2 a x 4 a x 2 a = b (126)
    tanh a x d x = 1 a ln cosh a x (127)
    e a x tanh b x d x = e ( a + 2 b ) x ( a + 2 b ) ( 2 F 1 ) 1 + a 2 b , 1 , 2 + a 2 b , e 2 b x e a x a ( 2 F 1 ) 1 , a 2 b , 1 + a 2 b , e 2 b x a b e a x 2 tan 1 [ e a x ] a a = b (128)
    cos a x cosh b x d x = 1 a 2 + b 2 a sin a x cosh b x + b cos a x sinh b x (129)
    cos a x sinh b x d x = 1 a 2 + b 2 b cos a x cosh b x + a sin a x sinh b x (130)
    sin a x cosh b x d x = 1 a 2 + b 2 a cos a x cosh b x + b sin a x sinh b x (131)
    sin a x sinh b x d x = 1 a 2 + b 2 b cosh b x sin a x a cos a x sinh b x (132)
    sinh a x cosh a x d x = 1 4 a 2 a x + sinh 2 a x (133)
    sinh a x cosh b x d x = 1 b 2 a 2 b cosh b x sinh a x a cosh a x sinh b x (134)

    
    Home | list | wiki | about | donate | index | forum | help | chat | translated from Chinese | 中文