TITLE: Differentiation wrt parameter $\int_0^\infty \sin^2(x)\cdot(x^2(x^2+1))^{-1}dx$
QUESTION [18 upvotes]: Use differentiation with respect to parameter obtaining a differential equation to solve
$$
\int_0^\infty \frac{\sin^2(x)}{x^2(x^2+1)}dx
$$
No complex variables, only this approach.  Interesting integral and it should have a nice ODE.  I have not found the right way yet.  we have singularities at $x=\pm i$.

REPLY [3 votes]: How about this method? Denote the integral by $I$. Let
$$ I(a,b)=\int_0^\infty \frac{\sin(ax)\sin(bx)}{x^2(1+x^2)}dx. $$
Then $I(0,b)=I(a,0)=0$, $I(1,1)=I$ and
\begin{eqnarray}
\frac{\partial^2I}{\partial a\partial b}&=&\int_0^\infty\frac{\cos(ax)\cos(bx)}{1+x^2}dx\\
&=&\frac{1}{2}\int_0^\infty\frac{\cos((a+b)x)}{1+x^2}dx+\frac{1}{2}\int_0^\infty\frac{\cos((a-b)x)}{1+x^2}dx\\
&=&\frac{\pi}{4}(e^{-(a+b)}+e^{-|a-b|})
\end{eqnarray}
where we use
$$ \int_0^\infty\frac{\cos(\alpha x)}{1+x^2}dx=\frac{\pi}{2}e^{-|\alpha|}. $$
Thus
$$ I=\frac{\pi}{4}\int_0^1\int_0^1(e^{-(a+b)}+e^{-|a-b|})dadb=\frac{(1+e^2)\pi}{4e^2}. $$