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Zeeshan 01

Member

Registered: 2016-07-22

Posts: 746

Rationalization

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

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Malik

Zeeshan 01

Member

Registered: 2016-07-22

Posts: 746

Re: Rationalization

I perform rationalization and ans is wrong

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Malik

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Rationalization

What did you get?

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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.

Bob

Administrator

Registered: 2010-06-20

Posts: 10,626

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

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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

zetafunc

Moderator

Registered: 2014-05-21

Posts: 2,436

Website

Re: Rationalization

You can use the tangent half-angle substitution.

Zeeshan 01

Member

Registered: 2016-07-22

Posts: 746

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)

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Malik

zetafunc

Moderator

Registered: 2014-05-21

Posts: 2,436

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Re: Rationalization

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.

Zeeshan 01

Member

Registered: 2016-07-22

Posts: 746

Re: Rationalization

Ok

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Malik

Zeeshan 01

Member

Registered: 2016-07-22

Posts: 746

Re: Rationalization

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

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Malik

zetafunc

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Registered: 2014-05-21

Posts: 2,436

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Re: Rationalization

You are missing something in the last line.

Zeeshan 01

Member

Registered: 2016-07-22

Posts: 746

Re: Rationalization

1÷sin^2 (x) -cos (x)÷sin^2 (x)

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Malik

Zeeshan 01

Member

Registered: 2016-07-22

Posts: 746

Re: Rationalization

I solve ans is cosecx-cot x+c

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Malik

Bob

Administrator

Registered: 2010-06-20

Posts: 10,626

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. 

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

zetafunc

Moderator

Registered: 2014-05-21

Posts: 2,436

Website

Re: Rationalization

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

Zeeshan 01

Member

Registered: 2016-07-22

Posts: 746

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???

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Malik

greg1313

Member

Registered: 2016-12-19

Posts: 20

Re: Rationalization

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

Zeeshan 01

Member

Registered: 2016-07-22

Posts: 746

Re: Rationalization

I know

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Malik

Bob

Administrator

Registered: 2010-06-20

Posts: 10,626

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.

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

So we have the same answer. 

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

Zeeshan 01

Member

Registered: 2016-07-22

Posts: 746

Re: Rationalization

Ok no one can wrong my method !!!

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Malik

Bob

Administrator

Registered: 2010-06-20

Posts: 10,626

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

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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

Zeeshan 01

Member

Registered: 2016-07-22

Posts: 746

Re: Rationalization

I know !!!! But thx

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

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Malik

zetafunc

Moderator

Registered: 2014-05-21

Posts: 2,436

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Re: Rationalization

Try completing the square and then using a trig substitution.

Zeeshan 01

Member

Registered: 2016-07-22

Posts: 746

Re: Rationalization

What is trig substitution

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

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Malik

zetafunc

Moderator

Registered: 2014-05-21

Posts: 2,436

Website

Re: Rationalization

Use the fact that

.

Zeeshan 01

Member

Registered: 2016-07-22

Posts: 746

Re: Rationalization

Plese show me

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Malik