Double integrals that involve integration by parts

In summary, the conversation discusses evaluating a double integral using the integral by parts method. The individual has attempted to solve the integral but is stuck at the x integration. They have shared pictures of their work and have tried breaking the last integration into three parts, but still cannot find a solution. The answer to the question is provided and the individual realizes that it would have been easier to do the integral with respect to x first.
  • #1
jwxie
282
0

Homework Statement



Evaluate the double integral.

Homework Equations



integral by part: uv - integral of v*du

The Attempt at a Solution



I wrote out my works. And I am just stuck at the x integration...

Please click on the links to see the pictures. I don't want to resize it which may reduce the quality.

http://i786.photobucket.com/albums/yy145/gokoproject/100702_150423.jpg
http://i786.photobucket.com/albums/yy145/gokoproject/100702_150440.jpg

I tried to break the last integration into three parts, which each had x^2 at the bottom. But then the integration by parts seem impossible...

The answer to the question is 19/2 - 1/2e^6
I am sure the first part with respect to y is correct (I verified that with wolframalpha).Any help would be appreciate. Thank you

* I believe doing dx first is easier, if I am correct. But I already started, so I just want to see how can i crack down with dy first. There can be ugly expression during exam too.
 
Last edited:
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  • #2
Doing it dx first is a LOT easier. If you do it dy first you need incomplete gamma functions to write down an indefinite integral (as you are discovering). There's no elementary way (like integration by parts) to do that integral.
 
  • #3
oh god. lol
thanks Dick. I see what you mean :)
 

FAQ: Double integrals that involve integration by parts

What is the purpose of using integration by parts in double integrals?

Integration by parts is a technique used to simplify the integration of a product of two functions. In the context of double integrals, it can help reduce the complexity of the integrand and make it easier to evaluate.

How do you determine which function to differentiate and which function to integrate in integration by parts for double integrals?

In general, it is recommended to differentiate the more complicated function and integrate the simpler one. However, in double integrals, it is also important to consider the symmetry of the integrand and choose the functions accordingly.

Can integration by parts be applied to all types of double integrals?

No, integration by parts can only be used for double integrals where the integrand can be expressed as a product of two functions. In cases where this is not possible, other integration techniques must be used.

Are there any special cases or exceptions when using integration by parts for double integrals?

Yes, there are certain cases where integration by parts may not work or may not be the most efficient method. These include integrands with infinite boundaries, singularities, or discontinuities.

Is there a general formula or rule for evaluating double integrals using integration by parts?

No, there is no one-size-fits-all formula for integration by parts in double integrals. The approach and choice of functions may vary depending on the specific integrand and the desired outcome. Practice and familiarity with different techniques are key to effectively using integration by parts.

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