Can one show that, for every $a$ and $b$ in $\mathbb R$,
$$ \int_{0}^{\infty} \! \frac {1}{1+x^{2}} \frac {x^{a}-x^{b}}{(1+x^{a})(1+x^{b})}~\mathrm{d}x=0\ ? $$
Any hints?
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is it $x^{x^{b}}$ – Jun 05 '11 at 11:06
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@Chandru: No, it is just $(1+x^b)$. – night owl Jun 05 '11 at 11:27
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@Rasholnikov: Right, but I think it wants it for any general $(a,b) \in \mathbb{R}$. – night owl Jun 05 '11 at 11:29
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Yes, one can. Here are some hints, which should be expanded before being called a proof.
Writing $x^a-x^b$ as $(x^a+1)-(x^b+1)$ and simplifying the fraction, one sees that it is enough to show that $I(a)$ does not depend on $a$, with $$ I(a)=\int_0^{+\infty}\frac{\mathrm{d}x}{(1+x^2)(1+x^a)} $$ To prove this, one could decompose $I(a)$ as the sum of an integral from $0$ to $1$ and an integral from $1$ to $+\infty$ and use the change of variable $y\leftarrow1/x$ in the latter. One would be left with $$ I(a)=\int_0^{1}\frac{\mathrm{d}x}{(1+x^2)(1+x^a)}+\int_0^{1}\frac{y^a\mathrm{d}y}{(1+y^2)(1+y^a)}=\int_0^{1}\frac{\mathrm{d}x}{1+x^2}, $$ which is independent on $a$, and this would yield the result.
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