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#1 2006-03-30 09:05:51
- MathsIsFun
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Integrals
Integrals
"The physicists defer only to mathematicians, and the mathematicians defer only to God ..." - Leon M. Lederman
#2 2006-04-07 01:05:19
- ganesh
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Re: Integrals
Standard Integrals of elementary functions
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#3 2006-04-07 01:56:35
- ganesh
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Re: Integrals
Derivative of indefinite integral, integral of derivative
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#4 2006-04-10 02:57:46
- mikau
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Re: Integrals
Isn't the integeral of 1/x dx supposed to contain absolute value symbols? ln |x|?
A logarithm is just a misspelled algorithm.
#5 2006-04-19 11:50:00
- George,Y
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Re: Integrals
Parallel to Post 3, we have Rule in Leibniz's notations
d∫=nothing, or you can delete them together
∫d=nothing, but you should add C at the end
Leibniz claimed his notations (d∫)and using them to form rules such as
d(uv)=udv+vdu could simplify the algebra. So they maybe an alternative for you.
X'(y-Xβ)=0
#6 2006-04-22 16:17:42
- ganesh
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Re: Integrals
Some important integrals
The integration constant c has been omitted.
Character is who you are when no one is looking.
#7 2006-04-22 16:37:17
- ganesh
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Re: Integrals
Important forms of Integrals
The integration constant c has been omitted.
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#8 2006-04-22 17:28:05
- ganesh
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Re: Integrals
Integrals of Logarithmic functions
The integration constant c has been omitted.
where
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#9 2006-04-22 17:37:40
- ganesh
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Re: Integrals
Integrals of Inverse Trignometric Functions
(The integration constant c has been omitted)
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#10 2006-04-22 17:44:03
- Ricky
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Re: Integrals
"In the real world, this would be a problem. But in mathematics, we can just define a place where this problem doesn't exist. So we'll go ahead and do that now..."
#11 2006-04-27 00:14:18
- ganesh
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Re: Integrals
Definite Integrals
Properties of Definite Integral
If then
If then
If f(x) is an even function, that is f(-x)=f(x), then
If f(x) is an odd function, that is f(-x)=-f(x), then
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#12 2006-04-27 01:17:34
- ganesh
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Re: Integrals
Area under curves
The area bounded by the curve y=f(x), x=a, x=b and the abcissa (x-axis) is
Similarly, the area bounded by the curve x=f(y), y=c, y=d and the ordiante (y-axis) is
Area between two curves
The area of the region bounded by the curves y=f(x) and y=g(x) and the lines x=a and x=b where f and g are continuous functions and f(x)≥g(x) for all x in [a,b] is
Character is who you are when no one is looking.
#13 2006-05-02 02:10:16
- ganesh
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Re: Integrals
Partial fractions
Form of the rational function Form of the partial fraction
where
cannot be factored further.
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#14 2006-05-03 02:03:02
- ganesh
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Re: Integrals
Example for using partial fraction method in Integration
The integrand can be rewritten as
or
Let
By solving for A and B, we get A=-5, B=10.
Therefore,
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#15 2006-05-06 22:41:52
- ganesh
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Re: Integrals
Integrals of Hyperbolic functions
The integration constant c, to be added on the Right Hand Side, has been omitted.
Integrals of Inverse Hyperbolic Functions
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#16 2006-05-07 16:33:14
- ganesh
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Re: Integrals
Bernoulli's formula for integration
If u', u'', u''' etc denote the first, second, third derivatives of the function u and v1, v2, v3 etc are the successive integrals of the function v, then
Example
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#17 2006-05-07 23:16:59
- ganesh
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Re: Integrals
Integrals of functions of the from x²±a²
The integration constant c, to be added on the Right Hand Side, has been omitted.
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#18 2006-08-06 17:10:39
- Zhylliolom
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Re: Integrals
Arc Length
The length of a curve y = f(x) from x = a to x = b is given by
If the curve is represented parametrically by x = f(t) and y = g(t), then the length of the curve from t = a to t = b is given by
In polar coordinates with r = f(θ), the length of the curve from θ = α to θ = β is given by
Volumes of Revolution
Disk method:
Washer method:
Shell method:
Iterated Integrals
If the double integral of f(x, y) over a region R bounded by f1(x) ≤ y ≤ f2(x), a ≤ x ≤ b exists, then we may write
This may be extended to triple integrals and beyond.
Transformations of Multiple Integrals
If (u, v) are the curvilinear coordinates of a point related to Cartesian coordinates by the transformation equations x = f(u, v), y = g(u, v) which map the region R to R' and G(u, v) = F(f(u, v), g(u, v)) then
This may be extended to triple integrals and beyond.
Note: See the section on Jacobians in the Partial Differentiation Formulas thread if you do not understand the notation used in "Transformations of Multiple Integrals":
http://www.mathsisfun.com/forum/viewtop … 823#p33823