Integrate 3cos^2(x): Step by Step Solution

[ZaZu]

One last thing, Umm I want to ask another question that does not really need a new thread.

In integration by parts, which function has the priority to be substituted as U

I follow the method of LITE
L = Logs and Ln
I = Indices
T = Trig Functions
E = Exponents

But integrating e^x.Cos(x)

The answer booklet chose e^x as U ... :S :S why is that ?

Which one has a priority to be replaced in U when we have a Trig function and an exponent function

 

[Cyosis]

It really doesn't matter which one you take in this case and don't forget LITE is a rule of thumb and it is not always the best course of action.

 

[ZaZu]

I tried taking the trig function for U
but the answer sheet gives a different answer, it used u = exponential function

What way do you recommend ?

 

[Hootenanny]

Personally, I wouldn't use integration by parts to evaluate the integral of e^xcos(x). If you know Euler's formula you can transform the cosine into the real part of a complex exponential, the integration then becomes straightforward.

 

[ZaZu]

Hmmm, I really do not know Euler's formula, my exam is tomorrow so I don't think I would be very safe learning it today, might mix up tomorrow.

So other than Euler's formula, is using LITE good enough ? Our tutor told us to use LITE, so perhaps in the exam ill just indicate that I am using THAT method, the one they taught us. In this case I should not lose marks (I hope) because its been told by them to use LITE method.

If you have a better method other than Euler's formula, please tell me what it is.

Thank you very much, really appreciated.

 

[Hootenanny]

Hmmm, I really do not know Euler's formula, my exam is tomorrow so I don't think I would be very safe learning it today, might mix up tomorrow.

So other than Euler's formula, is using LITE good enough ? Our tutor told us to use LITE, so perhaps in the exam ill just indicate that I am using THAT method, the one they taught us. In this case I should not lose marks (I hope) because its been told by them to use LITE method.

If you have a better method other than Euler's formula, please tell me what it is.

Thank you very much, really appreciated.

The LITE method is fine.

 

[ZaZu]

Hmm so I should ignore what this answer sheet is telling me, because its following a different method other than LITE.
Its substituting e for U instead of cosx ..

Thanks :)

 

[Cyosis]

I get the same answer regardless whether I use u=e^x or u=cos x. I reckon you made an error with some minus signs, it's the only error I can really think of in this problem.

Whether you ignore the sheet or not you should get the same answer that is : $$\frac{1}{2} e^x(\cos x+\sin x)$$

 

[ZaZu]

Hmm I think I do have some errors in the signs.

The answer in the sheet is :

http://img149.imageshack.us/img149/7731/image357f.jpg

So you're right, using whatever way its the same answer !

 

[ephedyn]

I don't know what's this LITE thing, and never had to use it. I look at it this way:

There are 4 forms of products of functions:

(1) Only one function is integrable

a. function with another function that cannot be integrated (e.g. sin(ln x), tan^-1(x) ln, lg etc.)
--> differentiate the non-integrable function, integrate the other function
--> e.g. x^3 ln(x), x tan^-1 (x)

b. a constant with a function that cannot be integrated
--> differentiate the non-integrable function, integrate the constant
--> e.g. ln x = 1 * ln(x), sin(ln x) = 1 * sin(ln x)

(2) Both functions are integrable

a. only one turns into a constant after a few differentiations, the other does not
--> differentiate the one that turns into a constant, integrate the other
--> repeat integration by parts until reduced to constant
--> e.g. x * sin(x), x^2 * e^(2x)

b. neither turns into a constant after a few differentiations
--> differentiate either of the two, integrate the other
--> shift the newly generated integral once it takes the same form as the given integral to the left hand side of the equation, then divide both sides
--> e.g. e^x * sin x, e^(2x) * cos 2x

So the the one you've asked falls under the last category. You can choose to integrate either one of them. But I usually apply Euler's formula for the one you've asked (search google for "complexifying the integral") because the rule of the thumb is to use the easiest/fastest method - and integration by parts tends not to be. But don't bother learning the use of Euler's formula here now if you haven't done an introduction to complex numbers.

Instead, I think you might benefit if you tried integrating the 4x2 examples that I've given above so that you can get used to the various forms.

 

[ZaZu]

I just checked back on this topic !

Thank you very much ephedyn, that was really helpful !
Thank you ! REally appreciated :) :)

 

[ephedyn]

Mmhmm, no problem. Are you in high school? If you need any more help in math or physics you could send me a private message here, I'll try to help where I can.