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

MathsIsFun
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Registered: 2005-01-21
Posts: 7,711

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

Jai Ganesh
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Posts: 45,954

Re: Integrals

Standard Integrals of elementary functions


It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

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

Jai Ganesh
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Registered: 2005-06-28
Posts: 45,954

Re: Integrals

Derivative of indefinite integral, integral of derivative


It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

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: 1,379

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

Jai Ganesh
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Posts: 45,954

Re: Integrals

Some important integrals

The integration constant c has been omitted.


It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

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

Jai Ganesh
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Posts: 45,954

Re: Integrals

Important forms of Integrals

The integration constant c has been omitted.




It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

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

Jai Ganesh
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Posts: 45,954

Re: Integrals

Integrals of Logarithmic functions

The integration constant c has been omitted.


where


It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

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

Jai Ganesh
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Registered: 2005-06-28
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Re: Integrals

Integrals of Inverse Trignometric Functions

(The integration constant c has been omitted)


It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

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

Jai Ganesh
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Registered: 2005-06-28
Posts: 45,954

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 appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

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

Jai Ganesh
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Registered: 2005-06-28
Posts: 45,954

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 appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

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

Jai Ganesh
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Registered: 2005-06-28
Posts: 45,954

Re: Integrals

Partial fractions

Form of the rational function  Form of the partial fraction



where
cannot be factored further.


It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

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

Jai Ganesh
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Registered: 2005-06-28
Posts: 45,954

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 appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

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

Jai Ganesh
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Registered: 2005-06-28
Posts: 45,954

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 appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

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

Jai Ganesh
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Registered: 2005-06-28
Posts: 45,954

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 appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

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

Jai Ganesh
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Registered: 2005-06-28
Posts: 45,954

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 appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

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
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Registered: 2010-06-20
Posts: 10,052

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
Sometimes I deliberately make mistakes, just to test you!  …………….Bob smile

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

Nehushtan
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Registered: 2013-03-09
Posts: 957

Re: Integrals

bob bundy wrote:

Where did that come from?

Differentiation.


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

Bob
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Registered: 2010-06-20
Posts: 10,052

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
Sometimes I deliberately make mistakes, just to test you!  …………….Bob smile

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

Nehushtan
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Registered: 2013-03-09
Posts: 957

Re: Integrals


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

Bob
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Registered: 2010-06-20
Posts: 10,052

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
Sometimes I deliberately make mistakes, just to test you!  …………….Bob smile

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

Nehushtan
Member
Registered: 2013-03-09
Posts: 957

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 (2014-03-13 02:49:49)


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