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How does one integrate $\int xf(ax)f(bx) dx$?

I think it cries out for integration by parts, but I don't know how to split the integrand.


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For the example $f(x) = \mathrm e^{x^4}$ your integral reduces to $\int \mathrm e^{(a^4+b^4)x^2}\,dx$ which cannot be written through the elementary functions. Maybe there is $f'$ somewhere in the integrand? – Ilya Sep 27 '11 at 10:15
Depending on what you want it in terms of, there may be no general formula, through by-parts or otherwise. – anon Sep 27 '11 at 10:21
@Gortaur: Thanks. If I change it a little bit say $\int_0^1 xf(ax)f(bx) dx$, would it be better? I think I have seen it equal something like $af'(ax)f(bx)-bf'(bx)f(ax)\over b^2-a^2$ somewhere, but I am nit sure how to get to that. – Eddie Sep 27 '11 at 10:44
@anon: Thanks. What if I change it into a definite integral as stated in my reply to Gortaur's comment? – Eddie Sep 27 '11 at 10:45
My comment applies whether or not it's definite. The formula you wrote obviously doesn't work because it doesn't even involve $\int f dx$ in any capacity; make $f$ a constant function and you'll see a concrete discrepancy. – anon Sep 27 '11 at 10:48

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