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## #1 2006-03-29 10:05:51

MathsIsFun
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Registered: 2005-01-21
Posts: 7,664

### Integrals

Integrals

"The physicists defer only to mathematicians, and the mathematicians defer only to God ..."  - Leon M. Lederman

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## #2 2006-04-06 03:05:19

ganesh
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Registered: 2005-06-28
Posts: 23,303

### Re: Integrals

Standard Integrals of elementary functions

It is no good to try to stop knowledge from going forward. Ignorance is never better than knowledge - Enrico Fermi.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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## #3 2006-04-06 03:56:35

ganesh
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Registered: 2005-06-28
Posts: 23,303

### Re: Integrals

Derivative of indefinite integral, integral of derivative

It is no good to try to stop knowledge from going forward. Ignorance is never better than knowledge - Enrico Fermi.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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## #4 2006-04-09 04:57:46

mikau
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Registered: 2005-08-22
Posts: 1,504

### 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.

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## #5 2006-04-18 13:50:00

George,Y
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Registered: 2006-03-12
Posts: 1,306

### 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

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## #6 2006-04-21 18:17:42

ganesh
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Registered: 2005-06-28
Posts: 23,303

### Re: Integrals

Some important integrals

The integration constant c has been omitted.

It is no good to try to stop knowledge from going forward. Ignorance is never better than knowledge - Enrico Fermi.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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## #7 2006-04-21 18:37:17

ganesh
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Registered: 2005-06-28
Posts: 23,303

### Re: Integrals

Important forms of Integrals

The integration constant c has been omitted.

It is no good to try to stop knowledge from going forward. Ignorance is never better than knowledge - Enrico Fermi.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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## #8 2006-04-21 19:28:05

ganesh
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Registered: 2005-06-28
Posts: 23,303

### Re: Integrals

Integrals of Logarithmic functions

The integration constant c has been omitted.

where

It is no good to try to stop knowledge from going forward. Ignorance is never better than knowledge - Enrico Fermi.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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## #9 2006-04-21 19:37:40

ganesh
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Registered: 2005-06-28
Posts: 23,303

### Re: Integrals

Integrals of Inverse Trignometric Functions

(The integration constant c has been omitted)

It is no good to try to stop knowledge from going forward. Ignorance is never better than knowledge - Enrico Fermi.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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## #10 2006-04-21 19:44:03

Ricky
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Registered: 2005-12-04
Posts: 3,791

### 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..."

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## #11 2006-04-26 02:14:18

ganesh
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Registered: 2005-06-28
Posts: 23,303

### 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

It is no good to try to stop knowledge from going forward. Ignorance is never better than knowledge - Enrico Fermi.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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## #12 2006-04-26 03:17:34

ganesh
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Registered: 2005-06-28
Posts: 23,303

### 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

It is no good to try to stop knowledge from going forward. Ignorance is never better than knowledge - Enrico Fermi.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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## #13 2006-05-01 04:10:16

ganesh
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Registered: 2005-06-28
Posts: 23,303

### Re: Integrals

Partial fractions

Form of the rational function  Form of the partial fraction

where
cannot be factored further.

It is no good to try to stop knowledge from going forward. Ignorance is never better than knowledge - Enrico Fermi.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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## #14 2006-05-02 04:03:02

ganesh
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Registered: 2005-06-28
Posts: 23,303

### 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,

It is no good to try to stop knowledge from going forward. Ignorance is never better than knowledge - Enrico Fermi.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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## #15 2006-05-06 00:41:52

ganesh
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Registered: 2005-06-28
Posts: 23,303

### 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

It is no good to try to stop knowledge from going forward. Ignorance is never better than knowledge - Enrico Fermi.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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## #16 2006-05-06 18:33:14

ganesh
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Registered: 2005-06-28
Posts: 23,303

### 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

It is no good to try to stop knowledge from going forward. Ignorance is never better than knowledge - Enrico Fermi.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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## #17 2006-05-07 01:16:59

ganesh
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Registered: 2005-06-28
Posts: 23,303

### 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.

It is no good to try to stop knowledge from going forward. Ignorance is never better than knowledge - Enrico Fermi.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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## #18 2006-08-05 19:10:39

Zhylliolom
Real Member
Registered: 2005-09-05
Posts: 412

### 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 f[sub]1[/sub](x) ≤ y ≤ f[sub]2[/sub](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

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## #19 2014-03-12 02:09:58

gourish
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Registered: 2013-05-28
Posts: 153

### 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

"The man was just too bored so he invented maths for fun"
-some wise guy

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## #20 2014-03-12 21:54:38

bob bundy
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Registered: 2010-06-20
Posts: 8,139

### Re: Integrals

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

??

Where did that come from?

Bob

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei

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## #21 2014-03-12 22:52:10

Nehushtan
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Registered: 2013-03-09
Posts: 905
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### Re: Integrals

bob bundy wrote:

Where did that come from?

Differentiation.

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## #22 2014-03-13 00:33:42

bob bundy
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Registered: 2010-06-20
Posts: 8,139

### Re: Integrals

Let

then

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

Bob

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei

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## #23 2014-03-13 00:38:58

Nehushtan
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Registered: 2013-03-09
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### Re: Integrals

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## #24 2014-03-13 00:45:28

bob bundy
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Registered: 2010-06-20
Posts: 8,139

### Re: Integrals

I agree with you but

gourish wrote:

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

so he was integrating not differentiating.

Bob

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei

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## #25 2014-03-13 02:48:09

Nehushtan
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Registered: 2013-03-09
Posts: 905
Website

### Re: Integrals

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

Last edited by Nehushtan (2014-03-13 02:49:49)

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