Struggling with Integration: e^x \sin(\pi x)

In summary, the conversation discusses the integration of the expression \int_{}^{} e^x \sin(\pi x) dx using the method of integration by parts. The participants also mention an alternative approach using complex exponentials.
  • #1
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Ok, here is the integral i seem to be having some issues with. I know there's a very simple step I am missing.

[tex]\int_{}^{} e^x \sin(\pi x) dx [/tex]

i attempted to do this using by parts integration.

I tried u = [tex] \sin(\pi x) [/tex] so du= [tex] \pi \cos(\pi x) dx [/tex]
so then dv= [tex] e^x dx [/tex] and v= [tex] e^x [/tex]

after using [tex] uv- \int v du [/tex] I seem to be going in circles...can someone help?
 
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  • #2
Do integration by parts on [itex]\int v du [/itex]. You'll end up with

[tex]\int_{}^{} e^x \sin(\pi x) dx = something - \int_{}^{} e^x \sin(\pi x) dx [/tex]

Aaah.. add [itex]\int_{}^{} e^x \sin(\pi x) dx [/itex] on both sides. Divide by two. ta-dam.
 
  • #3
quasar987 said:
Divide by two.

Small warning, it won't look quite like you've described, the constant won't be two.
 
  • #4
Oh right.. because of the pies!
 
  • #5
There's a quick alternative way using complex exponentials.

Since
[tex]e^x \sin (\pi x)= \Im (e^{x+i\pi x})=\Im (e^{x(1+i\pi)})[/tex]

[tex]\int e^{x(1+i\pi)} dx=\frac{e^{x(1+i\pi)}}{1+i\pi}=\frac{(1-i\pi)e^{x(1+i\pi)}}{1+\pi^2}[/tex]

Now take the imaginary part.
 

1. What is the purpose of integrating e^x sin(πx)?

The purpose of integrating e^x sin(πx) is to find the total area under the curve of the function. This could be useful in various mathematical and scientific applications, such as calculating work done by a varying force or determining the displacement of a moving object.

2. How do you solve the integral of e^x sin(πx)?

To solve the integral of e^x sin(πx), you can use the integration by parts method. This involves breaking down the integral into two parts and using the product rule to solve for the integral. Alternatively, you can use substitution by letting u = sin(πx) and du = πcos(πx)dx, which simplifies the integral to e^x sin(πx)dx = (1/π)e^x du.

3. Can you explain the concept of integration by parts?

Integration by parts is a method of solving integrals that involves breaking down the integral into two parts and using the product rule to solve for the integral. The product rule states that the integral of the product of two functions is equal to the first function times the integral of the second function, minus the integral of the derivative of the first function times the integral of the second function.

4. Is there a shortcut or trick to solving the integral of e^x sin(πx)?

Unfortunately, there is no shortcut or trick to solving the integral of e^x sin(πx). It requires the use of integration techniques such as integration by parts or substitution. Practice and familiarity with these techniques can make the process easier and quicker, but there is no quick fix for solving integrals.

5. What are some real-life applications of integrating e^x sin(πx)?

Integrating e^x sin(πx) can be used in various real-life applications, such as calculating the work done by a varying force, determining the displacement of a moving object, or finding the total amount of energy used over a period of time. It can also be applied in engineering, physics, and economics to solve various problems involving rates of change and accumulation.

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