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<center><math> \int_{0}^{\infty}(e^{-x^2}*cos(2x)) dx</math></center> | <center><math> \int_{0}^{\infty}(e^{-x^2}*cos(2x)) dx</math></center> | ||
As we can see, there isn't any particular place that we can use u-substitution or integration by parts to produce a solution easily, but Feynman shows us how we can parameterize the integral as a function, focusing on the cosine factor of the integrand. By writing the integral as a function, we can change the expression to: | As we can see, there isn't any particular place that we can use u-substitution or integration by parts to produce a solution easily, but Feynman shows us how we can parameterize the integral as a function, focusing on the cosine factor of the integrand. By writing the integral as a function, we can change the expression to: | ||
| − | <center><math> F(a | + | <center><math> F(a) = \int_{0}^{\infty}(e^{-x^2}*cos(a*x)) dx</math> (where a = 2)</center> |
This allows us to extract an x from the cosine segment of the integrand by differentiating with respect to a, making the left portion of the integrand <math>x*e^{-x^2}</math>, which is much easier to deal with than just <math>e^{-x^2}</math> | This allows us to extract an x from the cosine segment of the integrand by differentiating with respect to a, making the left portion of the integrand <math>x*e^{-x^2}</math>, which is much easier to deal with than just <math>e^{-x^2}</math> | ||
| + | |||
| + | From here, our differentiated equation is <math> F'(a) = \int_{0}^{\infty}(-x*e^{-x^2}*sin{(a*x)}) dx</math>, which we can then integrate using integration by parts. | ||
[[ Walther MA271 Fall2020 topic14 | Back to Feynman Integrals]] | [[ Walther MA271 Fall2020 topic14 | Back to Feynman Integrals]] | ||
Revision as of 18:41, 27 November 2020
What is Feynman's Technique?
Feynman's Technique of integration utilizes parametrization and a mix of other different mathematical properties in order to integrate an integral that is can't be integrated through normal processes like u-substitution or integration by parts. It primarily focuses on setting a function equal to an integral, and then differentiating the function to get an integral that is easier to work with. A simple example would be an integral such as:
As we can see, there isn't any particular place that we can use u-substitution or integration by parts to produce a solution easily, but Feynman shows us how we can parameterize the integral as a function, focusing on the cosine factor of the integrand. By writing the integral as a function, we can change the expression to:
This allows us to extract an x from the cosine segment of the integrand by differentiating with respect to a, making the left portion of the integrand $ x*e^{-x^2} $, which is much easier to deal with than just $ e^{-x^2} $
From here, our differentiated equation is $ F'(a) = \int_{0}^{\infty}(-x*e^{-x^2}*sin{(a*x)}) dx $, which we can then integrate using integration by parts.
