Exponential integration confused

In summary, the conversation discusses how to integrate the function e^-x (sin(nπx)), with the individual trying different methods such as part integration and substitution. However, they encounter difficulties and the conversation suggests using a specific trick of integrating by parts twice and solving for the original integral. Further resources are also provided for a complete solution to the problem.
  • #1
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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  • #2
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.
 

1. What is exponential integration?

Exponential integration is a mathematical method used to solve integrals involving exponential functions. It is a technique that allows us to find the area under a curve of an exponential function.

2. How is exponential integration different from regular integration?

Exponential integration is different from regular integration because it specifically deals with exponential functions, while regular integration can be used for a wide range of functions. Exponential integration involves using specific rules and techniques that are unique to exponential functions.

3. What are some common examples of exponential integration?

Some common examples of exponential integration include finding the area under the curve of functions such as e^x, e^-x, and a^x, where a is a constant. These types of integrals can be found using techniques like integration by parts or substitution.

4. Why is exponential integration important?

Exponential integration is important because exponential functions are commonly found in many real-world applications, such as population growth, compound interest, and radioactive decay. Being able to solve integrals involving exponential functions allows us to analyze and understand these real-world phenomena.

5. What are some tips for solving exponential integration problems?

Some tips for solving exponential integration problems include identifying the type of exponential function being used, being familiar with integration rules for exponential functions, and using algebraic manipulations to simplify the integral before attempting to solve it. It is also important to practice and develop a good understanding of basic integration techniques.

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