Exponential integration confused

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SUMMARY

The integral of the function e-xsin(nπx) can be solved using integration by parts. By defining I = ∫ e-xsin(nπx) dx and applying integration by parts twice, the original integral reappears alongside additional terms. This allows for the formulation of an equation that can be solved for I. For a complete solution, refer to the resource provided at e-academia.cz.

PREREQUISITES
  • Understanding of integration techniques, specifically integration by parts.
  • Familiarity with exponential functions and trigonometric integrals.
  • Knowledge of substitution methods in calculus.
  • Basic algebra skills for solving equations.
NEXT STEPS
  • Study the method of integration by parts in detail.
  • Learn about solving differential equations involving exponential and trigonometric functions.
  • Explore advanced integration techniques, including reduction formulas.
  • Review resources on integral calculus, such as e-academia.cz for practical examples.
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Students and professionals in mathematics, particularly those studying calculus, as well as educators looking for effective methods to teach integration techniques.

JI567
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Hi, does anyone know how to integrate e^-x (sin(nπx))? I have tried part integration but that goes on until infinity... and I am not sure how to use the substitution method...Please help! I have tried taking e^-x as U but then I end up getting the entire canceled off then...
 
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Do you mean: $$\int e^{-x}\sin n\pi x\; dx$$

There's a trick - as you noticed, integration by parts gets you the original integral back again with some extra terms, so round and round you go.
The trick is this: put $$I = \int e^{-x}\sin n\pi x\; dx$$ ... integrate the RHS by parts, twice only.
When you've done that, you should see the original integral appear again, with a bunch of other stuff.
Replace the integral with "I" and solve the resulting equation for I.
 

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