Tabular Integration by Parts Repeated

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The discussion focuses on using the tabular method for integration by parts, particularly for integrals that repeat, such as ∫e^x sin(x) dx. It highlights that while traditional repeated integration by parts can lead to a solvable equation, the tabular method may appear to result in endless derivatives and integrals. However, it is noted that when a previously encountered form is reached, one can stop and solve for the integral. This approach helps organize the process and avoid unnecessary repetition. Ultimately, the tabular method remains a valid technique for such integrals despite the potential for repetition.
benjaug
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So, in my class we are learning how to use the tabular method to solve an integration by parts problem... but what happens if the two parts of the integral continuously repeat?
The example I have in mind is
\int e^x sin(x) dx.
I know how to solve this using repeated integration by parts... solve it until the integrand is e^x sinx again and then add to both sides and divide by 2... that makes sense to me. But when you use the tabular method for it, the derivatives of u and the integrals of v just continue endlessly... is it impossible to solve that way?

Thanks for the help!
 
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Tabular integration by parts is just a way to organize integration by parts when it may be repeated. In particular upon reaching a form previously encountered (without having merely undone a step) one can stop and solve for the integral.
 

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