Tabular Integration by Parts Repeated

[benjaug]

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!

 

[lurflurf]

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.