MHB How can I integral this problem?

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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.
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\[ \int dx_1dx_2...dx_d e^{(x^2_1+x^2_2+...+x^2_d)^{r/2}} = \frac{\pi ^{d/2}(d/r)!}{(d/2)!} \]

How can I derive this answer?
 
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Thread 'Proving that convexity implies second order derivative being positive'

There are probably loads of proofs of this online, but I do not want to cheat. Here is my attempt: Convexity says that $$f(\lambda a + (1-\lambda)b) \leq \lambda f(a) + (1-\lambda) f(b)$$ $$f(b + \lambda(a-b)) \leq f(b) + \lambda (f(a) - f(b))$$ We know from the intermediate value theorem that there exists a ##c \in (b,a)## such that $$\frac{f(a) - f(b)}{a-b} = f'(c).$$ Hence $$f(b + \lambda(a-b)) \leq f(b) + \lambda (a - b) f'(c))$$ $$\frac{f(b + \lambda(a-b)) - f(b)}{\lambda(a-b)}...
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