How to compute the following integral

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Homework Help Overview

The discussion revolves around computing a specific integral involving an exponential function with a quadratic expression in the exponent. The integral presented includes terms that complicate the computation, particularly the interaction between the variable \( x \) and a parameter \( x_0 \).

Discussion Character

  • Exploratory, Assumption checking, Mathematical reasoning

Approaches and Questions Raised

  • Participants explore factoring the exponent and consider substitutions to simplify the integral. There are questions about the appropriateness of proposed substitutions and the implications of those choices on the integral's complexity.

Discussion Status

Several participants have offered different approaches, including factoring and substitution, while others express uncertainty about the effectiveness of these methods. There is an acknowledgment that the integral may not be elementary, suggesting a productive exploration of its properties.

Contextual Notes

Participants note the potential complexity introduced by the term involving \( x_0 \) and the possibility of needing to use special functions, such as error functions, in the solution. The discussion reflects a learning environment where assumptions about the integral's nature are being questioned.

thenewbosco
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I was wondering how to compute the following integral:
\int xe^{\frac{-x^2+2x\cdot x_0-x_{0}^2}{2a^2}}dx

it would be quite simple if there was no term in the exponent x times x0 i think...any help on this thanks
 
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Factor the expression in the exponent and then make a substitution.
 
i factored the exponent to be -(x-x0)^2 but what is the substitution to make? u = x-xo? this does not help because of the x...
 
use integration by parts after making the substitution.
 
so the substitution i proposed is the correct one to make?
 
Try u=(x-x0)2, I think that should work, but it might be a bit messy, I'm pretty sure parts will not work though.
 
but for what you propose then du=2(x-xo)dx, which cannot be put into the integral like this...
 
It has to have some error function in it, so definitely it's not an elementary integral.

Daniel.
 

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