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**MathsIsFun****Administrator**- Registered: 2005-01-21
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Integrals

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

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**ganesh****Moderator**- Registered: 2005-06-28
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**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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**ganesh****Moderator**- Registered: 2005-06-28
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**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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**mikau****Member**- Registered: 2005-08-22
- Posts: 1,504

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

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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**ganesh****Moderator**- Registered: 2005-06-28
- Posts: 21,088

**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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**ganesh****Moderator**- Registered: 2005-06-28
- Posts: 21,088

**Important forms of Integrals**

The integration constant c has been omitted.

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

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**ganesh****Moderator**- Registered: 2005-06-28
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**Integrals of Logarithmic functions**

The integration constant c has been omitted.

where

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

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**ganesh****Moderator**- Registered: 2005-06-28
- Posts: 21,088

**Integrals of Inverse Trignometric Functions**

(The integration constant c has been omitted)

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

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

"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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**ganesh****Moderator**- Registered: 2005-06-28
- Posts: 21,088

**Definite Integrals**

**Properties of Definite Integral**

If

thenIf

thenIf 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

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

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**ganesh****Moderator**- Registered: 2005-06-28
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**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

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

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**ganesh****Moderator**- Registered: 2005-06-28
- Posts: 21,088

**Partial fractions**

** Form of the rational function Form of the partial fraction**

where

cannot be factored further.

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

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**ganesh****Moderator**- Registered: 2005-06-28
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**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,

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

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**ganesh****Moderator**- Registered: 2005-06-28
- Posts: 21,088

**Integrals of Hyperbolic functions**

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

**Integrals of Inverse Hyperbolic Functions**

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

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**ganesh****Moderator**- Registered: 2005-06-28
- Posts: 21,088

**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**

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

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**ganesh****Moderator**- Registered: 2005-06-28
- Posts: 21,088

**Integrals of functions of the from x²±a² **

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

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

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

**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":*

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**gourish****Member**- Registered: 2013-05-28
- Posts: 153

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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**bob bundy****Moderator**- Registered: 2010-06-20
- Posts: 7,613

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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**bob bundy****Moderator**- Registered: 2010-06-20
- Posts: 7,613

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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**bob bundy****Moderator**- Registered: 2010-06-20
- Posts: 7,613

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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