Rationalization

Zeeshan 01 · 2017-03-16 16:34:35

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

Zeeshan 01 · 2017-03-16 16:35:37

I perform rationalization and ans is wrong

bobbym · 2017-03-16 18:49:32

What did you get?

bob bundy · 2017-03-16 21:51:54

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

zetafunc · 2017-03-16 22:09:22

You can use the tangent half-angle substitution.

Zeeshan 01 · 2017-03-17 01:26:30

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)

zetafunc · 2017-03-17 01:55:55

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.

Zeeshan 01 · 2017-03-17 02:38:09

Ok

Zeeshan 01 · 2017-03-17 02:40:49

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

zetafunc · 2017-03-17 02:47:47

You are missing something in the last line.

Zeeshan 01 · 2017-03-17 04:02:17

1÷sin^2 (x) -cos (x)÷sin^2 (x)

Zeeshan 01 · 2017-03-17 04:41:42

I solve ans is cosecx-cot x+c

bob bundy · 2017-03-17 06:24:57

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

zetafunc · 2017-03-17 07:34:41

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

Zeeshan 01 · 2017-03-17 15:09:03

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???

greg1313 · 2017-03-17 15:58:35

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

Zeeshan 01 · 2017-03-17 20:12:46

I know

bob bundy · 2017-03-17 20:28:11

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

Zeeshan 01 · 2017-03-18 00:32:08

Ok no one can wrong my method !!!

bob bundy · 2017-03-18 00:38:33

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

Zeeshan 01 · 2017-03-18 05:32:47

I know !!!! But thx

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

zetafunc · 2017-03-18 05:57:27

Try completing the square and then using a trig substitution.

Zeeshan 01 · 2017-03-18 15:30:59

What is trig substitution

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

zetafunc · 2017-03-18 19:28:59

Use the fact that .

Zeeshan 01 · 2017-03-18 21:02:57

Plese show me

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

Rationalization

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

Re: Rationalization

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

Re: Rationalization

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

Re: Rationalization

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

Re: Rationalization

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

Re: Rationalization

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

Re: Rationalization

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

Re: Rationalization

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

Re: Rationalization

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

Re: Rationalization

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

Re: Rationalization

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

Re: Rationalization

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

Re: Rationalization

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

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

Re: Rationalization

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

Re: Rationalization

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

Re: Rationalization

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

Re: Rationalization

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

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

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#21 2017-03-18 05:32:47

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

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

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

Re: Rationalization

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

Re: Rationalization

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