Cyclical Integration by Parts, going round and round

In summary, the person is trying to integrate by parts but keeps getting different answers. They are missing something and need to plug in numbers to see if they match what Wolfram Alpha gives them.
  • #1
GeekPioneer
13
0

Homework Statement



Integrate By Parts (i.e. not using formulas)
∫e3xcos(2x)dx


The Attempt at a Solution


I keep going around in circles, I know at some point I should be able to subtract the original integral across the = and then divide out the coefficient and that's the final answer. But I keep on getting different answers.

If some one could show the proper u=x1 du=x2 and v=x2 dv=x4 and state how many times I will have to repeat this and what u=x1 du=x2 and v=x2 dv=x4 should be for each step in the serial progression that is integration by parts. Please and Thanks
 
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  • #2
You should be able to integrate by parts twice. Then there is a nice trick. Can you show us what you have tried so far?
 
  • #3
y=∫e3xcos(2x)dx where u=e3x du=3e3x & v=(1/2)sin(2x) dv=cos(2x)

After integrating by parts twice i get this

y=e3xsin(2x)/2 - e3xsin(2x)/2 +sin2xe3x/2 -∫e3xcos(2x)dx


the 2nd time I integrated by parts it was

u=sin2x du=2cos(2x)

v=(1/3)e3x dv=e3x
 
  • #4
OK good. Now for the trick.

You have defined

[tex]y = \int e^{3x} \cos(2x) dx[/tex]

Now look at the right hand side of the result you got. The same integral appears again. So using the definition of y, you have

y = other stuff - y

Can you see what to do now?
 
  • #5
Yeah I can see it but my answer didn't match up with what Wolfram gave.

they gave 1/13 e^(3 x) (3 cos(2 x)+2 sin(2 x))

And what am I missing why do your equations look so much...prettier?
 
  • #6
And what did you get for your answer?
It's possible your answer is still equivalent to what WA gives you.
 
  • #7
Well yes or coarse you silly goose. I guess i need to plug in some numbers and see if they work out. Ill report back shortly.
 
  • #8
Its not coming out the same, I don't get it?
 

What is Cyclical Integration by Parts?

Cyclical Integration by Parts is a mathematical technique used to solve integrals that involve products of functions. It is a variation of the Integration by Parts method, where the process is repeated in a cyclical manner until the integral is solved.

How does Cyclical Integration by Parts work?

The process of Cyclical Integration by Parts involves selecting two functions to integrate, applying the Integration by Parts formula, and then repeating the process on the resulting integral until the original integral is solved. This is done by choosing different functions to differentiate and integrate with each repetition, hence the term "going round and round".

When should I use Cyclical Integration by Parts?

Cyclical Integration by Parts is useful when the original integral cannot be solved by traditional methods, such as substitution or partial fractions. It is also helpful when the integrand is a product of multiple functions or when the integrand contains trigonometric functions.

What are the benefits of using Cyclical Integration by Parts?

Cyclical Integration by Parts allows for the integration of more complex functions that cannot be solved by other methods. It also provides a systematic approach to solving integrals and can be used to solve a wide range of integrals, making it a valuable tool for mathematicians and scientists.

Are there any limitations to Cyclical Integration by Parts?

While Cyclical Integration by Parts can be an effective method for solving integrals, it may not always provide a solution. In some cases, the integral may become more complicated with each repetition, making it difficult to solve. Additionally, this method may not be suitable for integrals with a large number of terms or when the functions involved are highly complex.

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