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#1 2017-03-16 16:34:35

Zeeshan 01
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Registered: 2016-07-22
Posts: 596

Rationalization

Evaluate
∫ (1÷(1+cos×(x))
Why we cannot rationalise it with 1-cosx


Math !!!!.

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#2 2017-03-16 16:35:37

Zeeshan 01
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Registered: 2016-07-22
Posts: 596

Re: Rationalization

I perform rationalization and ans is wrong


Math !!!!.

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#3 2017-03-16 18:49:32

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: Rationalization

What did you get?


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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#4 2017-03-16 21:51:54

bob bundy
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Registered: 2010-06-20
Posts: 8,071

Re: Rationalization

hi Zeeshan 01

What do you mean by rationalise here?

eg.

This is called rationalisation because the irrational denominator has been made into a rational.

But cosine(x) isn't an irrational.  ??

To do the integral you can make use of

This makes a function that is directly integrable.

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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#5 2017-03-16 22:09:22

zetafunc
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Re: Rationalization

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#6 2017-03-17 01:26:30

Zeeshan 01
Member
Registered: 2016-07-22
Posts: 596

Re: Rationalization

I know half angle but why we not rationaliz

How???
But cosine(x) isn't an irrational.  ??

Why we cannot multiply and divide by   1-cos (x)


Math !!!!.

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#7 2017-03-17 01:55:55

zetafunc
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Re: Rationalization

Zeeshan 01 wrote:

Why we cannot multiply and divide by   1-cos (x)

No one is saying you can't do that, you can do that if you want. Post your calculation here and tell us what you think about the integral.

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#8 2017-03-17 02:38:09

Zeeshan 01
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Registered: 2016-07-22
Posts: 596

Re: Rationalization

Ok


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#9 2017-03-17 02:40:49

Zeeshan 01
Member
Registered: 2016-07-22
Posts: 596

Re: Rationalization

By doing rationalization
1-cosx÷sin (x)^2
Then
1÷sin^2 (x) -cos (x)÷


Math !!!!.

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#10 2017-03-17 02:47:47

zetafunc
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Re: Rationalization

You are missing something in the last line.

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#11 2017-03-17 04:02:17

Zeeshan 01
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Registered: 2016-07-22
Posts: 596

Re: Rationalization

1÷sin^2 (x) -cos (x)÷sin^2 (x)


Math !!!!.

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#12 2017-03-17 04:41:42

Zeeshan 01
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Registered: 2016-07-22
Posts: 596

Re: Rationalization

I solve ans is cosecx-cot x+c


Math !!!!.

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#13 2017-03-17 06:24:57

bob bundy
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Registered: 2010-06-20
Posts: 8,071

Re: Rationalization

At first I thought this was not correct as my answer was different.  But both results have the same graph so then I checked the identity and we have the same.  smile

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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#14 2017-03-17 07:34:41

zetafunc
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Registered: 2014-05-21
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Re: Rationalization

I would recommend the tangent half-angle substitution for integrals of this type.

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#15 2017-03-17 15:09:03

Zeeshan 01
Member
Registered: 2016-07-22
Posts: 596

Re: Rationalization

I also know half angle but why not this

t both results have the same graph  ????? Where you plotted graph
Then I checked the identity and we have the same.  smile
Which identity???


Math !!!!.

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#16 2017-03-17 15:58:35

greg1313
Member
Registered: 2016-12-19
Posts: 14

Re: Rationalization

Zeeshan 01 wrote:

By doing rationalization
1-cosx÷sin (x)^2  . . . . . .  This isn't correct, because you didn't use grouping symbols.

[1 - cos(x)]÷[sin(x)]^2

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#17 2017-03-17 20:12:46

Zeeshan 01
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Registered: 2016-07-22
Posts: 596

Re: Rationalization

I know


Math !!!!.

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#18 2017-03-17 20:28:11

bob bundy
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Registered: 2010-06-20
Posts: 8,071

Re: Rationalization

hi Zeeshan 01

I did this integral a different way.

So my first thought when I saw your answer was ???

So I plotted the two functions together on a graph.  The second plot fitted exactly over the first.  To show this more clearly I have offset my graph by 0.3 along the axis so you can see both graphs.

97QN4hh.gif

Then I checked to see if I could prove they are the same by using identity methods:

So we have the same answer.  smile

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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#19 2017-03-18 00:32:08

Zeeshan 01
Member
Registered: 2016-07-22
Posts: 596

Re: Rationalization

Ok no one can wrong my method !!!


Math !!!!.

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#20 2017-03-18 00:38:33

bob bundy
Administrator
Registered: 2010-06-20
Posts: 8,071

Re: Rationalization

hi Zeeshan 01

Your method and answer are both good.  With any integration, if your answer differentiates back to the original question, then it is a good answer.

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 2017-03-18 05:32:47

Zeeshan 01
Member
Registered: 2016-07-22
Posts: 596

Re: Rationalization

I know !!!! But thx

And this question check this
∫ 1÷(x^2+4x+13)dx
How to solve this??


Math !!!!.

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#22 2017-03-18 05:57:27

zetafunc
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Registered: 2014-05-21
Posts: 1,987
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Re: Rationalization

Try completing the square and then using a trig substitution.

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#23 2017-03-18 15:30:59

Zeeshan 01
Member
Registered: 2016-07-22
Posts: 596

Re: Rationalization

What is trig substitution

When I do 1÷((x+2)^2+3^2)
It's tan formula


Math !!!!.

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#24 2017-03-18 19:28:59

zetafunc
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Registered: 2014-05-21
Posts: 1,987
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Re: Rationalization

Use the fact that
.

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#25 2017-03-18 21:02:57

Zeeshan 01
Member
Registered: 2016-07-22
Posts: 596

Re: Rationalization

Plese show me


Math !!!!.

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