Tabular Integration by Parts Repeated

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SUMMARY

The discussion focuses on the application of the tabular method for solving integration by parts, specifically for the integral \(\int e^x \sin(x) dx\). Participants confirm that while the tabular method organizes the process, it can lead to endless derivatives and integrals when the parts repeat. The solution involves recognizing when the integrand returns to a previously encountered form, allowing for the equation to be solved by adding both sides and dividing by 2. This method effectively simplifies the resolution of integrals that exhibit repetitive patterns.

PREREQUISITES
  • Understanding of integration by parts
  • Familiarity with the tabular method for integration
  • Knowledge of derivatives and integrals of exponential and trigonometric functions
  • Basic algebraic manipulation skills
NEXT STEPS
  • Study the tabular integration method in detail
  • Practice solving integrals involving exponential and trigonometric functions
  • Explore advanced integration techniques, including repeated integration by parts
  • Learn about convergence and divergence in integrals with repeating patterns
USEFUL FOR

Students learning calculus, mathematics educators, and anyone seeking to master integration techniques, particularly those involving repeated integration by parts.

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!
 
Physics news on Phys.org
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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