SlideShare ist ein Scribd-Unternehmen logo
1 von 14
Downloaden Sie, um offline zu lesen
Table of Basic Integrals
Basic Forms
(1) xn
dx =
1
n + 1
xn+1
, n = −1
(2)
1
x
dx = ln |x|
(3) udv = uv − vdu
(4)
1
ax + b
dx =
1
a
ln |ax + b|
Integrals of Rational Functions
(5)
1
(x + a)2
dx = −
1
x + a
(6) (x + a)n
dx =
(x + a)n+1
n + 1
, n = −1
(7) x(x + a)n
dx =
(x + a)n+1
((n + 1)x − a)
(n + 1)(n + 2)
(8)
1
1 + x2
dx = tan−1
x
(9)
1
a2 + x2
dx =
1
a
tan−1 x
a
1
(10)
x
a2 + x2
dx =
1
2
ln |a2
+ x2
|
(11)
x2
a2 + x2
dx = x − a tan−1 x
a
(12)
x3
a2 + x2
dx =
1
2
x2
−
1
2
a2
ln |a2
+ x2
|
(13)
1
ax2 + bx + c
dx =
2
√
4ac − b2
tan−1 2ax + b
√
4ac − b2
(14)
1
(x + a)(x + b)
dx =
1
b − a
ln
a + x
b + x
, a = b
(15)
x
(x + a)2
dx =
a
a + x
+ ln |a + x|
(16)
x
ax2 + bx + c
dx =
1
2a
ln |ax2
+bx+c|−
b
a
√
4ac − b2
tan−1 2ax + b
√
4ac − b2
Integrals with Roots
(17)
√
x − a dx =
2
3
(x − a)3/2
(18)
1
√
x ± a
dx = 2
√
x ± a
(19)
1
√
a − x
dx = −2
√
a − x
2
(20) x
√
x − a dx =



2a
3
(x − a)3/2
+ 2
5
(x − a)5/2
, or
2
3
x(x − a)3/2
− 4
15
(x − a)5/2
, or
2
15
(2a + 3x)(x − a)3/2
(21)
√
ax + b dx =
2b
3a
+
2x
3
√
ax + b
(22) (ax + b)3/2
dx =
2
5a
(ax + b)5/2
(23)
x
√
x ± a
dx =
2
3
(x 2a)
√
x ± a
(24)
x
a − x
dx = − x(a − x) − a tan−1 x(a − x)
x − a
(25)
x
a + x
dx = x(a + x) − a ln
√
x +
√
x + a
(26) x
√
ax + b dx =
2
15a2
(−2b2
+ abx + 3a2
x2
)
√
ax + b
(27)
x(ax + b) dx =
1
4a3/2
(2ax + b) ax(ax + b) − b2
ln a
√
x + a(ax + b)
(28)
x3(ax + b) dx =
b
12a
−
b2
8a2x
+
x
3
x3(ax + b)+
b3
8a5/2
ln a
√
x + a(ax + b)
(29)
√
x2 ± a2 dx =
1
2
x
√
x2 ± a2 ±
1
2
a2
ln x +
√
x2 ± a2
3
(30)
√
a2 − x2 dx =
1
2
x
√
a2 − x2 +
1
2
a2
tan−1 x
√
a2 − x2
(31) x
√
x2 ± a2 dx =
1
3
x2
± a2 3/2
(32)
1
√
x2 ± a2
dx = ln x +
√
x2 ± a2
(33)
1
√
a2 − x2
dx = sin−1 x
a
(34)
x
√
x2 ± a2
dx =
√
x2 ± a2
(35)
x
√
a2 − x2
dx = −
√
a2 − x2
(36)
x2
√
x2 ± a2
dx =
1
2
x
√
x2 ± a2
1
2
a2
ln x +
√
x2 ± a2
(37)
√
ax2 + bx + c dx =
b + 2ax
4a
√
ax2 + bx + c+
4ac − b2
8a3/2
ln 2ax + b + 2 a(ax2 + bx+c)
x
√
ax2 + bx + c dx =
1
48a5/2
2
√
a
√
ax2 + bx + c −3b2
+ 2abx + 8a(c + ax2
)
+3(b3
− 4abc) ln b + 2ax + 2
√
a
√
ax2 + bx + c
(38)
4
(39)
1
√
ax2 + bx + c
dx =
1
√
a
ln 2ax + b + 2 a(ax2 + bx + c)
(40)
x
√
ax2 + bx + c
dx =
1
a
√
ax2 + bx + c−
b
2a3/2
ln 2ax + b + 2 a(ax2 + bx + c)
(41)
dx
(a2 + x2)3/2
=
x
a2
√
a2 + x2
Integrals with Logarithms
(42) ln ax dx = x ln ax − x
(43) x ln x dx =
1
2
x2
ln x −
x2
4
(44) x2
ln x dx =
1
3
x3
ln x −
x3
9
(45) xn
ln x dx = xn+1 ln x
n + 1
−
1
(n + 1)2
, n = −1
(46)
ln ax
x
dx =
1
2
(ln ax)2
(47)
ln x
x2
dx = −
1
x
−
ln x
x
5
(48) ln(ax + b) dx = x +
b
a
ln(ax + b) − x, a = 0
(49) ln(x2
+ a2
) dx = x ln(x2
+ a2
) + 2a tan−1 x
a
− 2x
(50) ln(x2
− a2
) dx = x ln(x2
− a2
) + a ln
x + a
x − a
− 2x
(51)
ln ax2
+ bx + c dx =
1
a
√
4ac − b2 tan−1 2ax + b
√
4ac − b2
−2x+
b
2a
+ x ln ax2
+ bx + c
(52) x ln(ax + b) dx =
bx
2a
−
1
4
x2
+
1
2
x2
−
b2
a2
ln(ax + b)
(53) x ln a2
− b2
x2
dx = −
1
2
x2
+
1
2
x2
−
a2
b2
ln a2
− b2
x2
(54) (ln x)2
dx = 2x − 2x ln x + x(ln x)2
(55) (ln x)3
dx = −6x + x(ln x)3
− 3x(ln x)2
+ 6x ln x
(56) x(ln x)2
dx =
x2
4
+
1
2
x2
(ln x)2
−
1
2
x2
ln x
(57) x2
(ln x)2
dx =
2x3
27
+
1
3
x3
(ln x)2
−
2
9
x3
ln x
6
Integrals with Exponentials
(58) eax
dx =
1
a
eax
(59)
√
xeax
dx =
1
a
√
xeax
+
i
√
π
2a3/2
erf i
√
ax , where erf(x) =
2
√
π
x
0
e−t2
dt
(60) xex
dx = (x − 1)ex
(61) xeax
dx =
x
a
−
1
a2
eax
(62) x2
ex
dx = x2
− 2x + 2 ex
(63) x2
eax
dx =
x2
a
−
2x
a2
+
2
a3
eax
(64) x3
ex
dx = x3
− 3x2
+ 6x − 6 ex
(65) xn
eax
dx =
xn
eax
a
−
n
a
xn−1
eax
dx
(66) xn
eax
dx =
(−1)n
an+1
Γ[1 + n, −ax], where Γ(a, x) =
∞
x
ta−1
e−t
dt
(67) eax2
dx = −
i
√
π
2
√
a
erf ix
√
a
7
(68) e−ax2
dx =
√
π
2
√
a
erf x
√
a
(69) xe−ax2
dx = −
1
2a
e−ax2
(70) x2
e−ax2
dx =
1
4
π
a3
erf(x
√
a) −
x
2a
e−ax2
Integrals with Trigonometric Functions
(71) sin ax dx = −
1
a
cos ax
(72) sin2
ax dx =
x
2
−
sin 2ax
4a
(73) sin3
ax dx = −
3 cos ax
4a
+
cos 3ax
12a
(74) sinn
ax dx = −
1
a
cos ax 2F1
1
2
,
1 − n
2
,
3
2
, cos2
ax
(75) cos ax dx =
1
a
sin ax
(76) cos2
ax dx =
x
2
+
sin 2ax
4a
(77) cos3
axdx =
3 sin ax
4a
+
sin 3ax
12a
8
(78) cosp
axdx = −
1
a(1 + p)
cos1+p
ax × 2F1
1 + p
2
,
1
2
,
3 + p
2
, cos2
ax
(79) cos x sin x dx =
1
2
sin2
x + c1 = −
1
2
cos2
x + c2 = −
1
4
cos 2x + c3
(80) cos ax sin bx dx =
cos[(a − b)x]
2(a − b)
−
cos[(a + b)x]
2(a + b)
, a = b
(81) sin2
ax cos bx dx = −
sin[(2a − b)x]
4(2a − b)
+
sin bx
2b
−
sin[(2a + b)x]
4(2a + b)
(82) sin2
x cos x dx =
1
3
sin3
x
(83) cos2
ax sin bx dx =
cos[(2a − b)x]
4(2a − b)
−
cos bx
2b
−
cos[(2a + b)x]
4(2a + b)
(84) cos2
ax sin ax dx = −
1
3a
cos3
ax
(85)
sin2
ax cos2
bxdx =
x
4
−
sin 2ax
8a
−
sin[2(a − b)x]
16(a − b)
+
sin 2bx
8b
−
sin[2(a + b)x]
16(a + b)
(86) sin2
ax cos2
ax dx =
x
8
−
sin 4ax
32a
(87) tan ax dx = −
1
a
ln cos ax
9
(88) tan2
ax dx = −x +
1
a
tan ax
(89) tann
ax dx =
tann+1
ax
a(1 + n)
× 2F1
n + 1
2
, 1,
n + 3
2
, − tan2
ax
(90) tan3
axdx =
1
a
ln cos ax +
1
2a
sec2
ax
(91) sec x dx = ln | sec x + tan x| = 2 tanh−1
tan
x
2
(92) sec2
ax dx =
1
a
tan ax
(93) sec3
x dx =
1
2
sec x tan x +
1
2
ln | sec x + tan x|
(94) sec x tan x dx = sec x
(95) sec2
x tan x dx =
1
2
sec2
x
(96) secn
x tan x dx =
1
n
secn
x, n = 0
(97) csc x dx = ln tan
x
2
= ln | csc x − cot x| + C
10
(98) csc2
ax dx = −
1
a
cot ax
(99) csc3
x dx = −
1
2
cot x csc x +
1
2
ln | csc x − cot x|
(100) cscn
x cot x dx = −
1
n
cscn
x, n = 0
(101) sec x csc x dx = ln | tan x|
Products of Trigonometric Functions and Mono-
mials
(102) x cos x dx = cos x + x sin x
(103) x cos ax dx =
1
a2
cos ax +
x
a
sin ax
(104) x2
cos x dx = 2x cos x + x2
− 2 sin x
(105) x2
cos ax dx =
2x cos ax
a2
+
a2
x2
− 2
a3
sin ax
(106) xn
cos xdx = −
1
2
(i)n+1
[Γ(n + 1, −ix) + (−1)n
Γ(n + 1, ix)]
11
(107) xn
cos ax dx =
1
2
(ia)1−n
[(−1)n
Γ(n + 1, −iax) − Γ(n + 1, ixa)]
(108) x sin x dx = −x cos x + sin x
(109) x sin ax dx = −
x cos ax
a
+
sin ax
a2
(110) x2
sin x dx = 2 − x2
cos x + 2x sin x
(111) x2
sin ax dx =
2 − a2
x2
a3
cos ax +
2x sin ax
a2
(112) xn
sin x dx = −
1
2
(i)n
[Γ(n + 1, −ix) − (−1)n
Γ(n + 1, −ix)]
(113) x cos2
x dx =
x2
4
+
1
8
cos 2x +
1
4
x sin 2x
(114) x sin2
x dx =
x2
4
−
1
8
cos 2x −
1
4
x sin 2x
(115) x tan2
x dx = −
x2
2
+ ln cos x + x tan x
(116) x sec2
x dx = ln cos x + x tan x
12
Products of Trigonometric Functions and Ex-
ponentials
(117) ex
sin x dx =
1
2
ex
(sin x − cos x)
(118) ebx
sin ax dx =
1
a2 + b2
ebx
(b sin ax − a cos ax)
(119) ex
cos x dx =
1
2
ex
(sin x + cos x)
(120) ebx
cos ax dx =
1
a2 + b2
ebx
(a sin ax + b cos ax)
(121) xex
sin x dx =
1
2
ex
(cos x − x cos x + x sin x)
(122) xex
cos x dx =
1
2
ex
(x cos x − sin x + x sin x)
Integrals of Hyperbolic Functions
(123) cosh ax dx =
1
a
sinh ax
(124) eax
cosh bx dx =



eax
a2 − b2
[a cosh bx − b sinh bx] a = b
e2ax
4a
+
x
2
a = b
(125) sinh ax dx =
1
a
cosh ax
13
(126) eax
sinh bx dx =



eax
a2 − b2
[−b cosh bx + a sinh bx] a = b
e2ax
4a
−
x
2
a = b
(127) tanh axdx =
1
a
ln cosh ax
(128) eax
tanh bx dx =



e(a+2b)x
(a + 2b)
2F1 1 +
a
2b
, 1, 2 +
a
2b
, −e2bx
−
1
a
eax
2F1 1,
a
2b
, 1 +
a
2b
, −e2bx
a = b
eax
− 2 tan−1
[eax
]
a
a = b
(129) cos ax cosh bx dx =
1
a2 + b2
[a sin ax cosh bx + b cos ax sinh bx]
(130) cos ax sinh bx dx =
1
a2 + b2
[b cos ax cosh bx + a sin ax sinh bx]
(131) sin ax cosh bx dx =
1
a2 + b2
[−a cos ax cosh bx + b sin ax sinh bx]
(132) sin ax sinh bx dx =
1
a2 + b2
[b cosh bx sin ax − a cos ax sinh bx]
(133) sinh ax cosh axdx =
1
4a
[−2ax + sinh 2ax]
(134) sinh ax cosh bx dx =
1
b2 − a2
[b cosh bx sinh ax − a cosh ax sinh bx]
c 2014. From http://integral-table.com, last revised June 14, 2014. This mate-
rial is provided as is without warranty or representation about the accuracy, correctness or
suitability of this material for any purpose. This work is licensed under the Creative Com-
mons Attribution-Noncommercial-Share Alike 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.
14

Weitere ähnliche Inhalte

Was ist angesagt?

Signals and systems: part i solutions
Signals and systems: part i solutionsSignals and systems: part i solutions
Signals and systems: part i solutionsPatrickMumba7
 
MCQs Ordinary Differential Equations
MCQs Ordinary Differential EquationsMCQs Ordinary Differential Equations
MCQs Ordinary Differential EquationsDrDeepaChauhan
 
Persamaan Laplace Dalam Bnetuk Koordinat Polar
Persamaan Laplace Dalam Bnetuk Koordinat PolarPersamaan Laplace Dalam Bnetuk Koordinat Polar
Persamaan Laplace Dalam Bnetuk Koordinat PolarSimesterious TheMaster
 
Tutorial 3 mth 3201
Tutorial 3 mth 3201Tutorial 3 mth 3201
Tutorial 3 mth 3201Drradz Maths
 
Fisika Dasar Ardi Mawardi
Fisika Dasar Ardi MawardiFisika Dasar Ardi Mawardi
Fisika Dasar Ardi Mawardifirdayanti8
 
Bai tap nguyen ham tich phan
Bai tap nguyen ham tich phanBai tap nguyen ham tich phan
Bai tap nguyen ham tich phantramhuuduc
 
Chuyên đề phương trình chứa căn thức bookbooming
Chuyên đề phương trình chứa căn thức   bookboomingChuyên đề phương trình chứa căn thức   bookbooming
Chuyên đề phương trình chứa căn thức bookboomingThế Giới Tinh Hoa
 
Solution Manual : Chapter - 06 Application of the Definite Integral in Geomet...
Solution Manual : Chapter - 06 Application of the Definite Integral in Geomet...Solution Manual : Chapter - 06 Application of the Definite Integral in Geomet...
Solution Manual : Chapter - 06 Application of the Definite Integral in Geomet...Hareem Aslam
 
Bdt đánh giá trên biên nhìn vào điểm nút
Bdt đánh giá trên biên nhìn vào điểm nútBdt đánh giá trên biên nhìn vào điểm nút
Bdt đánh giá trên biên nhìn vào điểm nútThế Giới Tinh Hoa
 
Lesson 22: Quadratic Forms
Lesson 22: Quadratic FormsLesson 22: Quadratic Forms
Lesson 22: Quadratic FormsMatthew Leingang
 
ANALISIS RIIL 1 3.3 dan 3.4 ROBERT G BARTLE
ANALISIS RIIL 1 3.3 dan 3.4 ROBERT G BARTLEANALISIS RIIL 1 3.3 dan 3.4 ROBERT G BARTLE
ANALISIS RIIL 1 3.3 dan 3.4 ROBERT G BARTLEMuhammad Nur Chalim
 
Signals and Systems part 2 solutions
Signals and Systems part 2 solutions Signals and Systems part 2 solutions
Signals and Systems part 2 solutions PatrickMumba7
 
Tuyển tập 500 bài Bất Đẳng Thức cổ điển
Tuyển tập 500 bài Bất Đẳng Thức cổ điểnTuyển tập 500 bài Bất Đẳng Thức cổ điển
Tuyển tập 500 bài Bất Đẳng Thức cổ điểnNguyễn Việt Long
 
電路學 - [第八章] 磁耦合電路
電路學 - [第八章] 磁耦合電路電路學 - [第八章] 磁耦合電路
電路學 - [第八章] 磁耦合電路Simen Li
 
Tutorial 1 mth 3201
Tutorial 1 mth 3201Tutorial 1 mth 3201
Tutorial 1 mth 3201Drradz Maths
 
Persamaan Diferensial Biasa (PDB) Orde 1
Persamaan Diferensial Biasa (PDB) Orde 1Persamaan Diferensial Biasa (PDB) Orde 1
Persamaan Diferensial Biasa (PDB) Orde 1made dwika
 
Đáp án chính thức môn Toán - Khối B - Kỳ thi Đại học năm 2011
Đáp án chính thức môn Toán - Khối B - Kỳ thi Đại học năm 2011Đáp án chính thức môn Toán - Khối B - Kỳ thi Đại học năm 2011
Đáp án chính thức môn Toán - Khối B - Kỳ thi Đại học năm 2011dethinet
 

Was ist angesagt? (20)

Integral table
Integral tableIntegral table
Integral table
 
Signals and systems: part i solutions
Signals and systems: part i solutionsSignals and systems: part i solutions
Signals and systems: part i solutions
 
MCQs Ordinary Differential Equations
MCQs Ordinary Differential EquationsMCQs Ordinary Differential Equations
MCQs Ordinary Differential Equations
 
Persamaan Laplace Dalam Bnetuk Koordinat Polar
Persamaan Laplace Dalam Bnetuk Koordinat PolarPersamaan Laplace Dalam Bnetuk Koordinat Polar
Persamaan Laplace Dalam Bnetuk Koordinat Polar
 
Tutorial 3 mth 3201
Tutorial 3 mth 3201Tutorial 3 mth 3201
Tutorial 3 mth 3201
 
Fisika Dasar Ardi Mawardi
Fisika Dasar Ardi MawardiFisika Dasar Ardi Mawardi
Fisika Dasar Ardi Mawardi
 
Bai tap nguyen ham tich phan
Bai tap nguyen ham tich phanBai tap nguyen ham tich phan
Bai tap nguyen ham tich phan
 
Chuyên đề phương trình chứa căn thức bookbooming
Chuyên đề phương trình chứa căn thức   bookboomingChuyên đề phương trình chứa căn thức   bookbooming
Chuyên đề phương trình chứa căn thức bookbooming
 
Solution Manual : Chapter - 06 Application of the Definite Integral in Geomet...
Solution Manual : Chapter - 06 Application of the Definite Integral in Geomet...Solution Manual : Chapter - 06 Application of the Definite Integral in Geomet...
Solution Manual : Chapter - 06 Application of the Definite Integral in Geomet...
 
Bdt đánh giá trên biên nhìn vào điểm nút
Bdt đánh giá trên biên nhìn vào điểm nútBdt đánh giá trên biên nhìn vào điểm nút
Bdt đánh giá trên biên nhìn vào điểm nút
 
Lesson 22: Quadratic Forms
Lesson 22: Quadratic FormsLesson 22: Quadratic Forms
Lesson 22: Quadratic Forms
 
ANALISIS RIIL 1 3.3 dan 3.4 ROBERT G BARTLE
ANALISIS RIIL 1 3.3 dan 3.4 ROBERT G BARTLEANALISIS RIIL 1 3.3 dan 3.4 ROBERT G BARTLE
ANALISIS RIIL 1 3.3 dan 3.4 ROBERT G BARTLE
 
Integral table
Integral tableIntegral table
Integral table
 
Sifat sifat fungsi
Sifat sifat fungsiSifat sifat fungsi
Sifat sifat fungsi
 
Signals and Systems part 2 solutions
Signals and Systems part 2 solutions Signals and Systems part 2 solutions
Signals and Systems part 2 solutions
 
Tuyển tập 500 bài Bất Đẳng Thức cổ điển
Tuyển tập 500 bài Bất Đẳng Thức cổ điểnTuyển tập 500 bài Bất Đẳng Thức cổ điển
Tuyển tập 500 bài Bất Đẳng Thức cổ điển
 
電路學 - [第八章] 磁耦合電路
電路學 - [第八章] 磁耦合電路電路學 - [第八章] 磁耦合電路
電路學 - [第八章] 磁耦合電路
 
Tutorial 1 mth 3201
Tutorial 1 mth 3201Tutorial 1 mth 3201
Tutorial 1 mth 3201
 
Persamaan Diferensial Biasa (PDB) Orde 1
Persamaan Diferensial Biasa (PDB) Orde 1Persamaan Diferensial Biasa (PDB) Orde 1
Persamaan Diferensial Biasa (PDB) Orde 1
 
Đáp án chính thức môn Toán - Khối B - Kỳ thi Đại học năm 2011
Đáp án chính thức môn Toán - Khối B - Kỳ thi Đại học năm 2011Đáp án chính thức môn Toán - Khối B - Kỳ thi Đại học năm 2011
Đáp án chính thức môn Toán - Khối B - Kỳ thi Đại học năm 2011
 

Andere mochten auch

Andere mochten auch (7)

Guia de laboratorio_1_electronica_analogica_ii__12667__
Guia de laboratorio_1_electronica_analogica_ii__12667__Guia de laboratorio_1_electronica_analogica_ii__12667__
Guia de laboratorio_1_electronica_analogica_ii__12667__
 
Vumetro estereo
Vumetro estereoVumetro estereo
Vumetro estereo
 
Proyecto final AMPLIFICACIÓN Y ACONDICIONAMIENTO DE SEÑALES
Proyecto final AMPLIFICACIÓN Y ACONDICIONAMIENTO DE SEÑALESProyecto final AMPLIFICACIÓN Y ACONDICIONAMIENTO DE SEÑALES
Proyecto final AMPLIFICACIÓN Y ACONDICIONAMIENTO DE SEÑALES
 
Diseño de un amplificador con mezclador de 2 canales y vumetro
Diseño de un amplificador con mezclador de 2 canales y vumetroDiseño de un amplificador con mezclador de 2 canales y vumetro
Diseño de un amplificador con mezclador de 2 canales y vumetro
 
Clases de polarizaciones de los amplificadores
Clases de polarizaciones de los   amplificadoresClases de polarizaciones de los   amplificadores
Clases de polarizaciones de los amplificadores
 
Montajes básicos de los transistores BJT
Montajes básicos de los transistores BJTMontajes básicos de los transistores BJT
Montajes básicos de los transistores BJT
 
Amplificador estereo 250 watts
Amplificador estereo 250 wattsAmplificador estereo 250 watts
Amplificador estereo 250 watts
 

Ähnlich wie Integral table (20)

Integral table
Integral tableIntegral table
Integral table
 
Single page-integral-table
Single page-integral-tableSingle page-integral-table
Single page-integral-table
 
Single page-integral-table
Single page-integral-table Single page-integral-table
Single page-integral-table
 
Tablas de integrales
Tablas de integralesTablas de integrales
Tablas de integrales
 
Integrales
IntegralesIntegrales
Integrales
 
Tablaintegrales
TablaintegralesTablaintegrales
Tablaintegrales
 
Tablaintegrales
TablaintegralesTablaintegrales
Tablaintegrales
 
Tablas integrales
Tablas integralesTablas integrales
Tablas integrales
 
Tabla integrales
Tabla integralesTabla integrales
Tabla integrales
 
Integrales
Integrales Integrales
Integrales
 
12 cbse-maths-2014-solution set 1
12 cbse-maths-2014-solution set 1 12 cbse-maths-2014-solution set 1
12 cbse-maths-2014-solution set 1
 
Tabla de integrales.pdf
Tabla de integrales.pdfTabla de integrales.pdf
Tabla de integrales.pdf
 
integrales.pdf
integrales.pdfintegrales.pdf
integrales.pdf
 
Tabla de-integrales
Tabla de-integralesTabla de-integrales
Tabla de-integrales
 
Leidy rivadeneira deber_1
Leidy rivadeneira deber_1Leidy rivadeneira deber_1
Leidy rivadeneira deber_1
 
ejercicio 140 libro de baldor resuelto
ejercicio 140 libro de baldor resueltoejercicio 140 libro de baldor resuelto
ejercicio 140 libro de baldor resuelto
 
Basic m4-2-chapter1
Basic m4-2-chapter1Basic m4-2-chapter1
Basic m4-2-chapter1
 
Ch13 16
Ch13 16Ch13 16
Ch13 16
 
3d. Pedagogy of Mathematics (Part II) - Algebra (Ex 3.4)
3d. Pedagogy of Mathematics (Part II) - Algebra (Ex 3.4)3d. Pedagogy of Mathematics (Part II) - Algebra (Ex 3.4)
3d. Pedagogy of Mathematics (Part II) - Algebra (Ex 3.4)
 
ejercicio 130 libro de baldor
ejercicio 130 libro de baldorejercicio 130 libro de baldor
ejercicio 130 libro de baldor
 

Integral table

  • 1. Table of Basic Integrals Basic Forms (1) xn dx = 1 n + 1 xn+1 , n = −1 (2) 1 x dx = ln |x| (3) udv = uv − vdu (4) 1 ax + b dx = 1 a ln |ax + b| Integrals of Rational Functions (5) 1 (x + a)2 dx = − 1 x + a (6) (x + a)n dx = (x + a)n+1 n + 1 , n = −1 (7) x(x + a)n dx = (x + a)n+1 ((n + 1)x − a) (n + 1)(n + 2) (8) 1 1 + x2 dx = tan−1 x (9) 1 a2 + x2 dx = 1 a tan−1 x a 1
  • 2. (10) x a2 + x2 dx = 1 2 ln |a2 + x2 | (11) x2 a2 + x2 dx = x − a tan−1 x a (12) x3 a2 + x2 dx = 1 2 x2 − 1 2 a2 ln |a2 + x2 | (13) 1 ax2 + bx + c dx = 2 √ 4ac − b2 tan−1 2ax + b √ 4ac − b2 (14) 1 (x + a)(x + b) dx = 1 b − a ln a + x b + x , a = b (15) x (x + a)2 dx = a a + x + ln |a + x| (16) x ax2 + bx + c dx = 1 2a ln |ax2 +bx+c|− b a √ 4ac − b2 tan−1 2ax + b √ 4ac − b2 Integrals with Roots (17) √ x − a dx = 2 3 (x − a)3/2 (18) 1 √ x ± a dx = 2 √ x ± a (19) 1 √ a − x dx = −2 √ a − x 2
  • 3. (20) x √ x − a dx =    2a 3 (x − a)3/2 + 2 5 (x − a)5/2 , or 2 3 x(x − a)3/2 − 4 15 (x − a)5/2 , or 2 15 (2a + 3x)(x − a)3/2 (21) √ ax + b dx = 2b 3a + 2x 3 √ ax + b (22) (ax + b)3/2 dx = 2 5a (ax + b)5/2 (23) x √ x ± a dx = 2 3 (x 2a) √ x ± a (24) x a − x dx = − x(a − x) − a tan−1 x(a − x) x − a (25) x a + x dx = x(a + x) − a ln √ x + √ x + a (26) x √ ax + b dx = 2 15a2 (−2b2 + abx + 3a2 x2 ) √ ax + b (27) x(ax + b) dx = 1 4a3/2 (2ax + b) ax(ax + b) − b2 ln a √ x + a(ax + b) (28) x3(ax + b) dx = b 12a − b2 8a2x + x 3 x3(ax + b)+ b3 8a5/2 ln a √ x + a(ax + b) (29) √ x2 ± a2 dx = 1 2 x √ x2 ± a2 ± 1 2 a2 ln x + √ x2 ± a2 3
  • 4. (30) √ a2 − x2 dx = 1 2 x √ a2 − x2 + 1 2 a2 tan−1 x √ a2 − x2 (31) x √ x2 ± a2 dx = 1 3 x2 ± a2 3/2 (32) 1 √ x2 ± a2 dx = ln x + √ x2 ± a2 (33) 1 √ a2 − x2 dx = sin−1 x a (34) x √ x2 ± a2 dx = √ x2 ± a2 (35) x √ a2 − x2 dx = − √ a2 − x2 (36) x2 √ x2 ± a2 dx = 1 2 x √ x2 ± a2 1 2 a2 ln x + √ x2 ± a2 (37) √ ax2 + bx + c dx = b + 2ax 4a √ ax2 + bx + c+ 4ac − b2 8a3/2 ln 2ax + b + 2 a(ax2 + bx+c) x √ ax2 + bx + c dx = 1 48a5/2 2 √ a √ ax2 + bx + c −3b2 + 2abx + 8a(c + ax2 ) +3(b3 − 4abc) ln b + 2ax + 2 √ a √ ax2 + bx + c (38) 4
  • 5. (39) 1 √ ax2 + bx + c dx = 1 √ a ln 2ax + b + 2 a(ax2 + bx + c) (40) x √ ax2 + bx + c dx = 1 a √ ax2 + bx + c− b 2a3/2 ln 2ax + b + 2 a(ax2 + bx + c) (41) dx (a2 + x2)3/2 = x a2 √ a2 + x2 Integrals with Logarithms (42) ln ax dx = x ln ax − x (43) x ln x dx = 1 2 x2 ln x − x2 4 (44) x2 ln x dx = 1 3 x3 ln x − x3 9 (45) xn ln x dx = xn+1 ln x n + 1 − 1 (n + 1)2 , n = −1 (46) ln ax x dx = 1 2 (ln ax)2 (47) ln x x2 dx = − 1 x − ln x x 5
  • 6. (48) ln(ax + b) dx = x + b a ln(ax + b) − x, a = 0 (49) ln(x2 + a2 ) dx = x ln(x2 + a2 ) + 2a tan−1 x a − 2x (50) ln(x2 − a2 ) dx = x ln(x2 − a2 ) + a ln x + a x − a − 2x (51) ln ax2 + bx + c dx = 1 a √ 4ac − b2 tan−1 2ax + b √ 4ac − b2 −2x+ b 2a + x ln ax2 + bx + c (52) x ln(ax + b) dx = bx 2a − 1 4 x2 + 1 2 x2 − b2 a2 ln(ax + b) (53) x ln a2 − b2 x2 dx = − 1 2 x2 + 1 2 x2 − a2 b2 ln a2 − b2 x2 (54) (ln x)2 dx = 2x − 2x ln x + x(ln x)2 (55) (ln x)3 dx = −6x + x(ln x)3 − 3x(ln x)2 + 6x ln x (56) x(ln x)2 dx = x2 4 + 1 2 x2 (ln x)2 − 1 2 x2 ln x (57) x2 (ln x)2 dx = 2x3 27 + 1 3 x3 (ln x)2 − 2 9 x3 ln x 6
  • 7. Integrals with Exponentials (58) eax dx = 1 a eax (59) √ xeax dx = 1 a √ xeax + i √ π 2a3/2 erf i √ ax , where erf(x) = 2 √ π x 0 e−t2 dt (60) xex dx = (x − 1)ex (61) xeax dx = x a − 1 a2 eax (62) x2 ex dx = x2 − 2x + 2 ex (63) x2 eax dx = x2 a − 2x a2 + 2 a3 eax (64) x3 ex dx = x3 − 3x2 + 6x − 6 ex (65) xn eax dx = xn eax a − n a xn−1 eax dx (66) xn eax dx = (−1)n an+1 Γ[1 + n, −ax], where Γ(a, x) = ∞ x ta−1 e−t dt (67) eax2 dx = − i √ π 2 √ a erf ix √ a 7
  • 8. (68) e−ax2 dx = √ π 2 √ a erf x √ a (69) xe−ax2 dx = − 1 2a e−ax2 (70) x2 e−ax2 dx = 1 4 π a3 erf(x √ a) − x 2a e−ax2 Integrals with Trigonometric Functions (71) sin ax dx = − 1 a cos ax (72) sin2 ax dx = x 2 − sin 2ax 4a (73) sin3 ax dx = − 3 cos ax 4a + cos 3ax 12a (74) sinn ax dx = − 1 a cos ax 2F1 1 2 , 1 − n 2 , 3 2 , cos2 ax (75) cos ax dx = 1 a sin ax (76) cos2 ax dx = x 2 + sin 2ax 4a (77) cos3 axdx = 3 sin ax 4a + sin 3ax 12a 8
  • 9. (78) cosp axdx = − 1 a(1 + p) cos1+p ax × 2F1 1 + p 2 , 1 2 , 3 + p 2 , cos2 ax (79) cos x sin x dx = 1 2 sin2 x + c1 = − 1 2 cos2 x + c2 = − 1 4 cos 2x + c3 (80) cos ax sin bx dx = cos[(a − b)x] 2(a − b) − cos[(a + b)x] 2(a + b) , a = b (81) sin2 ax cos bx dx = − sin[(2a − b)x] 4(2a − b) + sin bx 2b − sin[(2a + b)x] 4(2a + b) (82) sin2 x cos x dx = 1 3 sin3 x (83) cos2 ax sin bx dx = cos[(2a − b)x] 4(2a − b) − cos bx 2b − cos[(2a + b)x] 4(2a + b) (84) cos2 ax sin ax dx = − 1 3a cos3 ax (85) sin2 ax cos2 bxdx = x 4 − sin 2ax 8a − sin[2(a − b)x] 16(a − b) + sin 2bx 8b − sin[2(a + b)x] 16(a + b) (86) sin2 ax cos2 ax dx = x 8 − sin 4ax 32a (87) tan ax dx = − 1 a ln cos ax 9
  • 10. (88) tan2 ax dx = −x + 1 a tan ax (89) tann ax dx = tann+1 ax a(1 + n) × 2F1 n + 1 2 , 1, n + 3 2 , − tan2 ax (90) tan3 axdx = 1 a ln cos ax + 1 2a sec2 ax (91) sec x dx = ln | sec x + tan x| = 2 tanh−1 tan x 2 (92) sec2 ax dx = 1 a tan ax (93) sec3 x dx = 1 2 sec x tan x + 1 2 ln | sec x + tan x| (94) sec x tan x dx = sec x (95) sec2 x tan x dx = 1 2 sec2 x (96) secn x tan x dx = 1 n secn x, n = 0 (97) csc x dx = ln tan x 2 = ln | csc x − cot x| + C 10
  • 11. (98) csc2 ax dx = − 1 a cot ax (99) csc3 x dx = − 1 2 cot x csc x + 1 2 ln | csc x − cot x| (100) cscn x cot x dx = − 1 n cscn x, n = 0 (101) sec x csc x dx = ln | tan x| Products of Trigonometric Functions and Mono- mials (102) x cos x dx = cos x + x sin x (103) x cos ax dx = 1 a2 cos ax + x a sin ax (104) x2 cos x dx = 2x cos x + x2 − 2 sin x (105) x2 cos ax dx = 2x cos ax a2 + a2 x2 − 2 a3 sin ax (106) xn cos xdx = − 1 2 (i)n+1 [Γ(n + 1, −ix) + (−1)n Γ(n + 1, ix)] 11
  • 12. (107) xn cos ax dx = 1 2 (ia)1−n [(−1)n Γ(n + 1, −iax) − Γ(n + 1, ixa)] (108) x sin x dx = −x cos x + sin x (109) x sin ax dx = − x cos ax a + sin ax a2 (110) x2 sin x dx = 2 − x2 cos x + 2x sin x (111) x2 sin ax dx = 2 − a2 x2 a3 cos ax + 2x sin ax a2 (112) xn sin x dx = − 1 2 (i)n [Γ(n + 1, −ix) − (−1)n Γ(n + 1, −ix)] (113) x cos2 x dx = x2 4 + 1 8 cos 2x + 1 4 x sin 2x (114) x sin2 x dx = x2 4 − 1 8 cos 2x − 1 4 x sin 2x (115) x tan2 x dx = − x2 2 + ln cos x + x tan x (116) x sec2 x dx = ln cos x + x tan x 12
  • 13. Products of Trigonometric Functions and Ex- ponentials (117) ex sin x dx = 1 2 ex (sin x − cos x) (118) ebx sin ax dx = 1 a2 + b2 ebx (b sin ax − a cos ax) (119) ex cos x dx = 1 2 ex (sin x + cos x) (120) ebx cos ax dx = 1 a2 + b2 ebx (a sin ax + b cos ax) (121) xex sin x dx = 1 2 ex (cos x − x cos x + x sin x) (122) xex cos x dx = 1 2 ex (x cos x − sin x + x sin x) Integrals of Hyperbolic Functions (123) cosh ax dx = 1 a sinh ax (124) eax cosh bx dx =    eax a2 − b2 [a cosh bx − b sinh bx] a = b e2ax 4a + x 2 a = b (125) sinh ax dx = 1 a cosh ax 13
  • 14. (126) eax sinh bx dx =    eax a2 − b2 [−b cosh bx + a sinh bx] a = b e2ax 4a − x 2 a = b (127) tanh axdx = 1 a ln cosh ax (128) eax tanh bx dx =    e(a+2b)x (a + 2b) 2F1 1 + a 2b , 1, 2 + a 2b , −e2bx − 1 a eax 2F1 1, a 2b , 1 + a 2b , −e2bx a = b eax − 2 tan−1 [eax ] a a = b (129) cos ax cosh bx dx = 1 a2 + b2 [a sin ax cosh bx + b cos ax sinh bx] (130) cos ax sinh bx dx = 1 a2 + b2 [b cos ax cosh bx + a sin ax sinh bx] (131) sin ax cosh bx dx = 1 a2 + b2 [−a cos ax cosh bx + b sin ax sinh bx] (132) sin ax sinh bx dx = 1 a2 + b2 [b cosh bx sin ax − a cos ax sinh bx] (133) sinh ax cosh axdx = 1 4a [−2ax + sinh 2ax] (134) sinh ax cosh bx dx = 1 b2 − a2 [b cosh bx sinh ax − a cosh ax sinh bx] c 2014. From http://integral-table.com, last revised June 14, 2014. This mate- rial is provided as is without warranty or representation about the accuracy, correctness or suitability of this material for any purpose. This work is licensed under the Creative Com- mons Attribution-Noncommercial-Share Alike 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. 14