- #1
Another1
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Question
\[ \int dx_1dx_2...dx_d e^{(x^2_1+x^2_2+...+x^2_d)^{r/2}} = \frac{\pi ^{d/2}(d/r)!}{(d/2)!} \]
How can I derive this answer?
MHB How can I integral this problem?
- Thread starter Another1
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The discussion focuses on deriving the integral of the function e^{(x^2_1+x^2_2+...+x^2_d)^{r/2}} over multiple dimensions. The proposed solution involves using techniques from multivariable calculus and properties of the gamma function. Participants highlight the importance of recognizing the symmetry in the integrand and applying polar coordinates for simplification. The relationship between factorials and gamma functions is also emphasized in the derivation process. Understanding these mathematical concepts is crucial for successfully integrating the given expression.
Question
\[ \int dx_1dx_2...dx_d e^{(x^2_1+x^2_2+...+x^2_d)^{r/2}} = \frac{\pi ^{d/2}(d/r)!}{(d/2)!} \]
How can I derive this answer?
- #2
HOI
- 921
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"Integral" is a noun. The verb is "integrate".
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