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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




































Character is who you are when no one is looking.
 

#3 2006-04-07 01:56:35

ganesh
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Re: Integrals

Derivative of indefinite integral, integral of derivative




Character is who you are when no one is looking.
 

#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.












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#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




Character is who you are when no one is looking.
 

#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 xa

The integration constant c, to be added on the Right Hand Side, has been omitted.



















Character is who you are when no one is looking.
 

#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

 

#19 2014-03-13 01:09:58

gourish
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Re: Integrals

integral of cot(x)= -cosec^2(x)+c but integral of cot(x)=log(sin(x))+c why do we have two results for the integration of the same function? @ganesh


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#20 2014-03-13 20:54:38

bob bundy
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Re: Integrals

integral of cot(x)= -cosec^2(x)+c

??

Where did that come from?

Bob


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#21 2014-03-13 21:52:10

Nehushtan
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Re: Integrals

bob bundy wrote:

Where did that come from?

Differentiation.


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#22 2014-03-13 23:33:42

bob bundy
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Re: Integrals

Let



then





As far as I can see this is not the same as cot(x).  ???

Bob


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#23 2014-03-13 23:38:58

Nehushtan
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Re: Integrals


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#24 2014-03-13 23:45:28

bob bundy
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Re: Integrals

I agree with you but

gourish wrote:

integral of cot(x)= -cosec^2(x)+c

so he was integrating not differentiating.

Bob


You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
 

#25 2014-03-14 01:48:09

Nehushtan
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Re: Integrals

Why do you still not get it? He clearly mistook the derivative of cot x for the integral. neutral

Last edited by Nehushtan (2014-03-14 01:49:49)


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