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MathsIsFun

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Registered: 2005-01-21

Posts: 7,713

Integrals

Integrals

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

Jai Ganesh

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Registered: 2005-06-28

Posts: 48,384

Re: Integrals

Standard Integrals of elementary functions

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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.

Jai Ganesh

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Posts: 48,384

Re: Integrals

Derivative of indefinite integral, integral of derivative

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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.

mikau

Member

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|?

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A logarithm is just a misspelled algorithm.

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.

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X'(y-Xβ)=0

Jai Ganesh

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Registered: 2005-06-28

Posts: 48,384

Re: Integrals

Some important integrals

The integration constant c has been omitted.

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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.

Jai Ganesh

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Registered: 2005-06-28

Posts: 48,384

Re: Integrals

Important forms of Integrals

The integration constant c has been omitted.

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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.

Jai Ganesh

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Registered: 2005-06-28

Posts: 48,384

Re: Integrals

Integrals of Logarithmic functions

The integration constant c has been omitted.

where

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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.

Jai Ganesh

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Registered: 2005-06-28

Posts: 48,384

Re: Integrals

Integrals of Inverse Trignometric Functions

(The integration constant c has been omitted)

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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.

Ricky

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Registered: 2005-12-04

Posts: 3,791

Re: Integrals

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"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..."

Jai Ganesh

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Registered: 2005-06-28

Posts: 48,384

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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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.

Jai Ganesh

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Registered: 2005-06-28

Posts: 48,384

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

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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.

Jai Ganesh

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Registered: 2005-06-28

Posts: 48,384

Re: Integrals

Partial fractions

Form of the rational function  Form of the partial fraction

where

cannot be factored further.

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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.

Jai Ganesh

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Registered: 2005-06-28

Posts: 48,384

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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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.

Jai Ganesh

Administrator

Registered: 2005-06-28

Posts: 48,384

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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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.

Jai Ganesh

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Registered: 2005-06-28

Posts: 48,384

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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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.

Jai Ganesh

Administrator

Registered: 2005-06-28

Posts: 48,384

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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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.

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

gourish

Member

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

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"The man was just too bored so he invented maths for fun"
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Bob

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Registered: 2010-06-20

Posts: 10,618

Re: Integrals

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

??

Where did that come from?

Bob

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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

Nehushtan

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Registered: 2013-03-09

Posts: 957

Re: Integrals

Where did that come from?

Differentiation.

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Bob

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Registered: 2010-06-20

Posts: 10,618

Re: Integrals

Let

then

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

Bob

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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

Nehushtan

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Registered: 2013-03-09

Posts: 957

Re: Integrals

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Bob

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Registered: 2010-06-20

Posts: 10,618

Re: Integrals

I agree with you but

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

so he was integrating not differentiating.

Bob

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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

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.

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

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