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 Comm
Trang 1Table of Integrals ∗ Basic Forms
Z
xndx = 1
n + 1x
n+1
(1) Z
1
Z udv = uv −
Z
Z
1
ax + bdx =
1
aln |ax + b| (4) Integrals of Rational Functions
Z 1 (x + a)2dx = − 1
Z
(x + a)ndx =(x + a)
n+1
n + 1 , n 6= −1 (6)
Z
x(x + a)ndx =(x + a)
n+1((n + 1)x − a) (n + 1)(n + 2) (7) Z
1
1 + x2dx = tan−1x (8) Z
1
a2+ x2dx =1
atan
−1x
Z
x
a2+ x2dx = 1
2ln |a
2
+ x2| (10)
Z
x2
a2+ x2dx = x − a tan−1x
Z
x3
a2+ x2dx =1
2x
2
−1 2
2
ln |a2+ x2| (12)
ax2+ bx + cdx =
2
√ 4ac − b2 tan−1√2ax + b
4ac − b2 (13)
Z
1
(x + a)(x + b)dx =
1
b − aln
a + x
b + x, a 6= b (14) Z
x (x + a)2dx = a
a + x+ ln |a + x| (15) Z
x
ax2+ bx + cdx =
1 2aln |ax
2
+ bx + c|
a√ 4ac − b2tan−1√2ax + b
4ac − b2 (16) Integrals with Roots
Z √
x − adx =2
3(x − a)
3/2
(17) Z
1
√
x ± adx = 2
√
√
a − xdx = −2
√
Z
x√
x − adx = 2
3a(x − a)
3/2
+2
5(x − a)
5/2
(20)
Z √
ax + bdx = 2b
3a+
2x 3
√
ax + b (21) Z
(ax + b)3/2dx = 2
5a(ax + b)
5/2
(22) Z
x
√
x ± adx =
2
3(x ∓ 2a)
√
x ± a (23)
Z r
x
a − xdx = −
p x(a − x) − a tan−1px(a − x)
x − a (24)
Z r
x
a + xdx =
p x(a + x) − a ln√x +√x + a (25)
Z
x√
ax + bdx = 2
15a2(−2b2+ abx + 3a2x2)√
ax + b (26)
Z p x(ax + b)dx = 1
4a3/2
h (2ax + b)pax(ax + b)
−b2ln
a√
x +pa(ax + b)
i (27)
Z p
x3(ax + b)dx =
b 12a− b
2
8a2x+
x 3
p
x3(ax + b)
+ b
3
8a5/2ln
a√
x +pa(ax + b)
(28)
Z p
x2± a2dx =1
2x
p
x2± a2±1
2
2
ln
x +px2± a2
(29)
Z p
a2− x2dx =1
2x
p
a2− x2+1
2
2
tan−1√ x
a2− x2
(30)
Z
xpx2± a2dx =1
3 x
2
± a23/2
(31)
Z 1
√
x2± a2dx = ln
x +px2± a2
Z 1
√
a2− x2dx = sin−1x
Z x
√
x2± a2dx =px2± a2 (34) Z
x
√
a2− x2dx = −pa2− x2 (35)
Z
x2
√
x2± a2dx =1
2x
p
x2± a2∓1
2
2
ln
x +px2± a2
(36)
Z p
ax2+ bx + cdx = b + 2ax
4a
p
ax2+ bx + c
+4ac − b
2
8a3/2 ln
2ax + b + 2pa(ax2+ bx+c)
(37)
Z
xpax2+ bx + c = 1
48a5/2
2√
apax2+ bx + c
× −3b2+ 2abx + 8a(c + ax2) +3(b3− 4abc) ln b + 2ax + 2√
apax2+ bx + c
(38)
Z
1
√
ax2+ bx + cdx =
1
√
aln 2ax + b + 2pa(ax2+ bx + c)
(39)
Z
x
√
ax2+ bx + cdx =
1 a
p
ax2+ bx + c
− b 2a3/2ln 2ax + b + 2pa(ax2+ bx + c)
(40)
Z dx (a2+ x2)3/2 = x
a2√
a2+ x2 (41)
Integrals with Logarithms
Z
ln axdx = x ln ax − x (42) Z
ln ax
x dx =
1
2(ln ax)
2
(43)
Z ln(ax + b)dx =
x +b a
ln(ax + b) − x, a 6= 0 (44)
Z ln(x2+ a2) dx = x ln(x2+ a2) + 2a tan−1x
a− 2x (45)
Z ln(x2− a2) dx = x ln(x2− a2) + a lnx + a
x − a− 2x (46)
Z
ln ax2+ bx + c dx = 1
a
p 4ac − b2tan−1√2ax + b
4ac − b2
− 2x +
b 2a+ x
ln ax2+ bx + c
(47)
Z
x ln(ax + b)dx = bx
2a−1
4x
2
+1 2
x2− b
2
a2
ln(ax + b) (48)
Z
x ln a2− b2x2 dx = −1
2x
2
+ 1
2
x2−a
2
b2
ln a2− b2
x2 (49)
Integrals with Exponentials
Z
eaxdx = 1
ae
ax
(50)
Z √
xeaxdx =1
a
√
xeax+ i
√ π 2a3/2erf i√
ax ,
where erf(x) =√2
π
Z x 0
e−t2dt (51)
Z
xexdx = (x − 1)ex (52) Z
xeaxdx = x
a − 1
a2
Z
x2exdx = x2− 2x + 2 ex
(54) Z
x2eaxdx = x2
a −2x
a2 + 2
a3
eax (55) Z
x3exdx = x3− 3x2+ 6x − 6 ex
(56) Z
xneaxdx =x
n
eax
a −n a
Z
xn−1eaxdx (57)
Z
xneaxdx = (−1)
n
an+1 Γ[1 + n, −ax], where Γ(a, x) =
Z ∞ x
ta−1e−tdt
(58)
Z
eax2dx = −i
√ π
2√
aerf ix
√
Z
e−ax2dx =
√ π
2√
aerf x
√
a
(60) Z
xe−ax2 dx = −1
2ae
−ax 2
(61)
Z
x2e−ax2 dx =1
4
r π
a3erf(x√
a) − x 2ae
−ax2
(62)
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1
Trang 2Integrals with Trigonometric Functions
Z
sin axdx = −1
Z
sin2axdx = x
2−sin 2ax
Z
sinnaxdx =
−1
acos ax 2F1
1
2,
1 − n
2 ,
3
2, cos
2
ax
(65)
Z
sin3axdx = −3 cos ax
4a +
cos 3ax
Z cos axdx =1
Z
cos2axdx =x
2+
sin 2ax
Z
cospaxdx = − 1
a(1 + p)cos
1+p
ax×
2F1
1 + p
2 ,
1
2,
3 + p
2 , cos
2
ax
(69)
Z
cos3axdx = 3 sin ax
4a +
sin 3ax
Z
cos ax sin bxdx = cos[(a − b)x]
2(a − b) −cos[(a + b)x]
2(a + b) , a 6= b
(71)
Z
sin2ax cos bxdx = −sin[(2a − b)x]
4(2a − b) +sin bx 2b −sin[(2a + b)x]
4(2a + b) (72) Z
sin2x cos xdx = 1
3sin
3
Z
cos2ax sin bxdx =cos[(2a − b)x]
4(2a − b) −cos bx
2b
−cos[(2a + b)x]
Z
cos2ax sin axdx = −1
3acos
3
Z
sin2ax cos2bxdx = x
4−sin 2ax 8a −sin[2(a − b)x]
16(a − b) +sin 2bx
8b −sin[2(a + b)x]
16(a + b) (76) Z
sin2ax cos2axdx =x
8 −sin 4ax
Z
tan axdx = −1
aln cos ax (78) Z
tan2axdx = −x +1
atan ax (79)
Z
tannaxdx =tan
n+1
ax a(1 + n) ×
2F1
n + 1
2 , 1,
n + 3
2 , − tan
2
ax
(80)
Z
tan3axdx =1
aln cos ax +
1 2asec
2
ax (81)
Z
sec xdx = ln | sec x + tan x| = 2 tanh−1tanx
2
(82)
Z
sec2axdx = 1
Z sec3x dx = 1
2sec x tan x +
1
2ln | sec x + tan x| (84) Z
sec x tan xdx = sec x (85) Z
sec2x tan xdx =1
2sec
2
Z secnx tan xdx = 1
nsec
n
x, n 6= 0 (87)
Z csc xdx = ln
tanx 2 = ln | csc x − cot x| + C (88)
Z csc2axdx = −1
acot ax (89) Z
csc3xdx = −1
2cot x csc x +
1
2ln | csc x − cot x| (90) Z
cscnx cot xdx = −1
ncsc
n
x, n 6= 0 (91) Z
sec x csc xdx = ln | tan x| (92)
Products of Trigonometric Functions and
Monomials
Z
x cos xdx = cos x + x sin x (93) Z
x cos axdx = 1
a2cos ax +x
asin ax (94) Z
x2cos xdx = 2x cos x + x2− 2 sin x (95)
Z
x2cos axdx =2x cos ax
a2 +a
2
x2− 2
a3 sin ax (96)
Z
xncosxdx = −1
2(i)
n+1
[Γ(n + 1, −ix) +(−1)nΓ(n + 1, ix)] (97)
Z
xncosaxdx = 1
2(ia)
1−n
[(−1)nΓ(n + 1, −iax)
Z
x sin xdx = −x cos x + sin x (99) Z
x sin axdx = −x cos ax
sin ax
a2 (100)
Z
x2sin xdx = 2 − x2 cos x + 2x sin x (101)
Z
x2sin axdx = 2 − a
2x2
a3 cos ax +2x sin ax
a2 (102)
Z
xnsin xdx = −1
2(i)
n
[Γ(n + 1, −ix) − (−1)nΓ(n + 1, −ix)]
(103) Products of Trigonometric Functions and
Exponentials
Z
exsin xdx =1
2e
x
(sin x − cos x) (104)
Z
ebxsin axdx = 1
a2+ b2ebx(b sin ax − a cos ax) (105)
Z
excos xdx =1
2e
x
(sin x + cos x) (106)
Z
ebxcos axdx = 1
a2+ b2ebx(a sin ax + b cos ax) (107)
Z
xexsin xdx =1
2e
x
(cos x − x cos x + x sin x) (108)
Z
xexcos xdx = 1
2e
x
(x cos x − sin x + x sin x) (109)
Integrals of Hyperbolic Functions
Z cosh axdx = 1
asinh ax (110)
Z
eaxcosh bxdx =
eax
a2− b2[a cosh bx − b sinh bx] a 6= b
e2ax 4a +
x
(111)
Z sinh axdx = 1
acosh ax (112)
Z
eaxsinh bxdx =
eax
a2− b2[−b cosh bx + a sinh bx] a 6= b
e2ax 4a −x
(113)
Z
eaxtanh bxdx =
e(a+2b)x
(a + 2b)2F1
h
1 + a 2b, 1, 2 +
a 2b, −e
2bxi
−1
ae
ax
2F1
ha 2b, 1, 1E, −e
2bxi
a 6= b
eax− 2 tan−1[eax]
(114)
Z tanh ax dx =1
aln cosh ax (115)
Z cos ax cosh bxdx = 1
a2+ b2[a sin ax cosh bx +b cos ax sinh bx] (116)
Z cos ax sinh bxdx = 1
a2+ b2[b cos ax cosh bx+
a sin ax sinh bx] (117)
Z sin ax cosh bxdx = 1
a2+ b2[−a cos ax cosh bx+
b sin ax sinh bx] (118)
Z sin ax sinh bxdx = 1
a2+ b2[b cosh bx sin ax−
a cos ax sinh bx] (119)
Z sinh ax cosh axdx = 1
4a[−2ax + sinh 2ax] (120)
Z sinh ax cosh bxdx = 1
b2− a2[b cosh bx sinh ax
−a cosh ax sinh bx] (121)
2