This document provides formulas for integrals of common functions including polynomials, rational functions, radicals, logarithms, exponentials, and trigonometric functions. It includes basic integral formulas like the integral of x^n dx as well as more complex integrals involving combinations of functions. There are over 100 formulas presented in a table format organized by function type.
A MAN’S RIGHT TO HAPPINESS If by happenstance you require emergency triage, Maybury recom- mends that you transform your image as much as possible to that of a health care provider.He means that doctors and nurses are social beings that will gravitate more quickly and attentively to those with whom they identify the most.Therefore,if you dress intelligently,sound highly educated, act very polite and courteous, and perhaps carry a book with you,health care providers may feel more inclined to pay more attention to you and provide better care. This is the kind of mindset you’ll need in the new Obamacare
CFD Simulation of By-pass Flow in a HRSG module by R&R Consult.pptxR&R Consult
CFD analysis is incredibly effective at solving mysteries and improving the performance of complex systems!
Here's a great example: At a large natural gas-fired power plant, where they use waste heat to generate steam and energy, they were puzzled that their boiler wasn't producing as much steam as expected.
R&R and Tetra Engineering Group Inc. were asked to solve the issue with reduced steam production.
An inspection had shown that a significant amount of hot flue gas was bypassing the boiler tubes, where the heat was supposed to be transferred.
R&R Consult conducted a CFD analysis, which revealed that 6.3% of the flue gas was bypassing the boiler tubes without transferring heat. The analysis also showed that the flue gas was instead being directed along the sides of the boiler and between the modules that were supposed to capture the heat. This was the cause of the reduced performance.
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It is always satisfying when we can help solve complex challenges like this. Do your systems also need a check-up or optimization? Give us a call!
Work done in cooperation with James Malloy and David Moelling from Tetra Engineering.
More examples of our work https://www.r-r-consult.dk/en/cases-en/
Student information management system project report ii.pdfKamal Acharya
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Water scarcity is the lack of fresh water resources to meet the standard water demand. There are two type of water scarcity. One is physical. The other is economic water scarcity.
Hierarchical Digital Twin of a Naval Power SystemKerry Sado
A hierarchical digital twin of a Naval DC power system has been developed and experimentally verified. Similar to other state-of-the-art digital twins, this technology creates a digital replica of the physical system executed in real-time or faster, which can modify hardware controls. However, its advantage stems from distributing computational efforts by utilizing a hierarchical structure composed of lower-level digital twin blocks and a higher-level system digital twin. Each digital twin block is associated with a physical subsystem of the hardware and communicates with a singular system digital twin, which creates a system-level response. By extracting information from each level of the hierarchy, power system controls of the hardware were reconfigured autonomously. This hierarchical digital twin development offers several advantages over other digital twins, particularly in the field of naval power systems. The hierarchical structure allows for greater computational efficiency and scalability while the ability to autonomously reconfigure hardware controls offers increased flexibility and responsiveness. The hierarchical decomposition and models utilized were well aligned with the physical twin, as indicated by the maximum deviations between the developed digital twin hierarchy and the hardware.
Welcome to WIPAC Monthly the magazine brought to you by the LinkedIn Group Water Industry Process Automation & Control.
In this month's edition, along with this month's industry news to celebrate the 13 years since the group was created we have articles including
A case study of the used of Advanced Process Control at the Wastewater Treatment works at Lleida in Spain
A look back on an article on smart wastewater networks in order to see how the industry has measured up in the interim around the adoption of Digital Transformation in the Water Industry.
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
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