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ggumdol
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Homework Statement
Can anybody prove the following double integral identity? How?:
[tex] \int_{0}^{1} s(1-s) f(sx) ds = \int_{0}^{1} s^2 \int_{0}^{1} t f(tsx) dt ds [/tex]
Here f(x) is an arbitrary Riemann-integrable function.
Thanks in advance.
Homework Equations
I've found the following but it does not seem to be applicable to my problem.
http://onlinelibrary.wiley.com/doi/10.1002/9783527618132.app3/pdf
The Attempt at a Solution
I tried Coordinate Transformation and Integration by Parts. I've working on this problem for a couple of days. Please help me. I guess this identity could be proved simply by a coordinate transformation.
For those who may wonder if this identity is really correct, please run the following Maple code, then you will get 0, implying that the above identity is correct.
[tex] f(x) = x^2 - e^{x} * cos(x) + e^{x^2}+x^2 (cos (x))^2 + x^3 + 7 x^{5/4} [/tex]
> f(x):=x^(2)-exp(x)*cos(x)+exp(x^(2))+x^(2)*(cos(x))^(2)+x^(3)+7*x^(5/(4));
> int(s*(1-s)*(eval(f(x), x = s*x)), s = 0 .. 1)-(int(s^2*(int(t*(eval(f(x), x = t*s*x)), t = 0 .. 1)), s = 0 .. 1));
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