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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.

Thanks.

Eddie
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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? – SBF Sep 27 '11 at 10:15
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    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
  • Typo: The $x$'s shouldn't be there after the integration in the refined example. – Eddie Sep 27 '11 at 10:51
  • There is certainly no way to evaluate the integral is terms of anything that brings us any closer to actually finding the integral. If we could work out this integral, then we could find the definite or indefinite integral of any function $g$ we desired simply by setting $a=b=1$, defining $ f = \sqrt{g/x} $ for definite integrals over $[0,1]$ and then scaling/translating the function for other intervals. – Ragib Zaman Sep 27 '11 at 11:00

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