HELP! Integration Using Euler's Formula (2 Viewers)

[kingkong123]

Hi,
Recently i've been looking at integrating using Euler's formula, i don't know when it will be useful but im trying to get the hang of it..

Ive been trying to do this integral:



using Euler's formula but i can't get it. I can do it using integration by parts, but was looking for a worked solution using Euler's.

To start i had
therefore

and im presuming you continue on my multiplying through by and integrate....

Can some one please help ))))))))

 

[Trebla]

 

[SpiralFlex]

Eulers form in my opinion is useful for integrating expressions that you often can't see directly. For this question you cannot integrate by parts or use any standard form unless you use double angles.

 

[kingkong123]

wow! thanks Trebla, ur a genius.
i'm yet to find a question where euler's formula is actually quicker than solving conventionally. can u shed some light?

cheers lad

 

[kingkong123]

Eulers form in my opinion is useful for integrating expressions that you often can't see directly. For this question you cannot integrate by parts or use any standard form unless you use double angles.

u can use integration by parts

 

[SpiralFlex]

u can use integration by parts

What I mean was you cannot always. If you have two completely obscure functions.

 

[math man]

The best method here would be to change Sind squared into cos2x form immediately and thus ibp would be quicker

 

[kingkong123]

The best method here would be to change Sind squared into cos2x form immediately and thus ibp would be quicker

this.

 

[kingkong123]

when would u use euler's formula? is it useful for ext2 hsc questions?

 

[math man]

But the question is still easy without changing sin squared, I just think the double angle substitution makes the question easier for every student

 

[math man]

when would u use euler's formula? is it useful for ext2 hsc questions?

Never, the method is beyond syllabus so you shouldn't really use I unless you can't do it another way

 

[kingkong123]

But the question is still easy without changing sin squared, I just think the double angle substitution makes the question easier for every student

changing to double angle results in an easier ibp. you'll use ibp to get the integral of e^xcos2x instead of integral of e^xsin^2x

 

[kingkong123]

Never, the method is beyond syllabus so you shouldn't really use I unless you can't do it another way

it doesn't matter if it's beyond the syllabus does it? if it works it works right? but will it ever be quicker :S

 

[IamBread]

it doesn't matter if it's beyond the syllabus does it? if it works it works right? but will it ever be quicker :S

I don't think you'll ever get a question where it'll be faster. MX2 integration isn't very hard.

 

[SpiralFlex]

Rational expressions prehaps.

 

[largarithmic]

I was under the impression 'beyond the syllabus' methods like calculusing complex wasnt allowed... I was told to completely avoid doing certain stuff I knew for inequalities and geometry, for instance.

In any case, can one of the uni guys explain to me the rigorous basis behind the calculus of complex numbers? I've seen it on occasion (such as when doing physics and using e^itheta to denote radial/tangential unit vectors) but Ive never had it explained or whatever. Do you just treat i as a constant or something?

 

[SpiralFlex]

I was under the impression 'beyond the syllabus' methods like calculusing complex wasnt allowed... I was told to completely avoid doing certain stuff I knew for inequalities and geometry, for instance.

In any case, can one of the uni guys explain to me the rigorous basis behind the calculus of complex numbers? I've seen it on occasion (such as when doing physics and using e^itheta to denote radial/tangential unit vectors) but Ive never had it explained or whatever. Do you just treat i as a constant or something?

My teacher said you could use anything you want as long as it is mathematically correct, but then again now that I think about it, it might be too dangerous.

 

[Trebla]

it doesn't matter if it's beyond the syllabus does it? if it works it works right? but will it ever be quicker :S

A good HSC question will always be designed such that the method within the scope of the syllabus is the best approach. Also, a general rule of thumb is that if you are going to use results that are not in the syllabus they will need to be derived. You are being assessed on how well you use the material within the syllabus, not how much research you did beyond it.

 

[Carrotsticks]

In any case, can one of the uni guys explain to me the rigorous basis behind the calculus of complex numbers? I've seen it on occasion (such as when doing physics and using e^itheta to denote radial/tangential unit vectors) but Ive never had it explained or whatever. Do you just treat i as a constant or something?

For simple cases, you are allowed to treat it as constant. When we take away 1, we only do so from the real component.

For example, the derivative of x^i is x^(-1+i). So the i remains untouched, but we -1 from the real component.

You explore deeper calculus in the complex field in the topic 'Complex Analysis' (2nd Year @ USYD).

Sub-topics that appear under this include:

- Contour integrals (you may have seen the symbol of the integral sign with a circle in the middle)
- Holomorphic functions

EDIT: You may find this interesting http://de2de.synechism.org/c7/sec72.pdf

My teacher said you could use anything you want as long as it is mathematically correct, but then again now that I think about it, it might be too dangerous.

The difficult thing is that most students cannot define what is mathematically correct or not. More often than not, the student proceeds with something that 'seems' correct, but it in fact lacks rigour.

Here is an example of a 'proof' for the fact that the Harmonic Series diverges. It may seem correct to the average student, but it is in fact extremely flawed.

 

[SpiralFlex]

For simple cases, you are allowed to treat it as constant. When we take away 1, we only do so from the real component.

For example, the derivative of x^i is x^(-1+i). So the i remains untouched, but we -1 from the real component.

You explore deeper calculus in the complex field in the topic 'Complex Analysis' (2nd Year @ USYD).

Sub-topics that appear under this include:

- Contour integrals (you may have seen the symbol of the integral sign with a circle in the middle)
- Holomorphic functions

EDIT: You may find this interesting http://de2de.synechism.org/c7/sec72.pdf



The difficult thing is that most students cannot define what is mathematically correct or not. More often than not, the student proceeds with something that 'seems' correct, but it in fact lacks rigour.

Here is an example of a 'proof' for the fact that the Harmonic Series diverges. It may seem correct to the average student, but it is in fact extremely flawed.

Define what you mean by "very flawed," I can only see a few "flaws". Also what program did you use?