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**Zeeshan 01****Member**- Registered: 2016-07-22
- Posts: 648

Evaluate

∫ (1÷(1+cos×(x))

Why we cannot rationalise it with 1-cosx

MZk

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**Zeeshan 01****Member**- Registered: 2016-07-22
- Posts: 648

I perform rationalization and ans is wrong

MZk

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

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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**bob bundy****Administrator**- Registered: 2010-06-20
- Posts: 8,232

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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You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei

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You can use the tangent half-angle substitution.

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**Zeeshan 01****Member**- Registered: 2016-07-22
- Posts: 648

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)

MZk

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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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**Zeeshan 01****Member**- Registered: 2016-07-22
- Posts: 648

Ok

MZk

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**Zeeshan 01****Member**- Registered: 2016-07-22
- Posts: 648

By doing rationalization

1-cosx÷sin (x)^2

Then

1÷sin^2 (x) -cos (x)÷

MZk

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You are missing something in the last line.

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**Zeeshan 01****Member**- Registered: 2016-07-22
- Posts: 648

1÷sin^2 (x) -cos (x)÷sin^2 (x)

MZk

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**Zeeshan 01****Member**- Registered: 2016-07-22
- Posts: 648

I solve ans is cosecx-cot x+c

MZk

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**bob bundy****Administrator**- Registered: 2010-06-20
- Posts: 8,232

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

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

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**Zeeshan 01****Member**- Registered: 2016-07-22
- Posts: 648

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???

MZk

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**greg1313****Member**- Registered: 2016-12-19
- Posts: 17

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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**Zeeshan 01****Member**- Registered: 2016-07-22
- Posts: 648

I know

MZk

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**bob bundy****Administrator**- Registered: 2010-06-20
- Posts: 8,232

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

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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**Zeeshan 01****Member**- Registered: 2016-07-22
- Posts: 648

Ok no one can wrong my method !!!

MZk

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**bob bundy****Administrator**- Registered: 2010-06-20
- Posts: 8,232

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

You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei

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**Zeeshan 01****Member**- Registered: 2016-07-22
- Posts: 648

I know !!!! But thx

And this question check this

∫ 1÷(x^2+4x+13)dx

How to solve this??

MZk

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Try completing the square and then using a trig substitution.

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**Zeeshan 01****Member**- Registered: 2016-07-22
- Posts: 648

What is trig substitution

When I do 1÷((x+2)^2+3^2)

It's tan formula

MZk

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Use the fact that .

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**Zeeshan 01****Member**- Registered: 2016-07-22
- Posts: 648

Plese show me

MZk

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