Recently I've been playing around with Feynman's Trick to evaluate integrals. Obviously, one of it's many great features is that it allows derivatives to make expressions simpler. I was wondering whether Laplace Transforms could equally be applied.
I'm not qualified to say the following is proper or rigorous, it was just an experiment.
Consider
$$I = \int_{0}^{\infty}\frac{\sin^2(x)}{x^2(x^2 + 1)}\, \mathrm dx.$$
Let
$$I(t) = \int_{0}^{\infty}\frac{\sin^2(tx)}{x^2(x^2 + 1)} \,\mathrm dx$$
Take the Laplace Transform to yield
\begin{align*}
\mathscr L[I(t)] &= \int_{0}^{\infty}\frac{\mathscr L[\sin^2(tx)]}{x^2(x^2 + 1)}\,\mathrm dx\\
&= \int_{0}^{\infty}\frac{\mathscr 1}{x^2(x^2 + 1)}\frac{2x^2}{s(s^2 + 4x^2)}\,\mathrm dx\\
&= \frac{2}{s}\int_{0}^{\infty}\frac{1}{(x^2 + 1)(4x^2 + s^2)}\,\mathrm dx.
\end{align*}
Splitting via Partial Fraction Decomposition we arrive at
\begin{align*}
\mathscr L[I(t)] &= \frac{2}{s(s^2 – 4)}\int_{0}^{\infty}\left[ \frac{1}{x^{2} + 1} – \frac{4}{4x^{2} + s^2}\right] \,\mathrm dx\\
&= \frac{2}{s(s^2 – 4)}\left[\arctan(x) – \frac{2}{s}\arctan\left(\frac{2x}{s}\right)\right]_{0}^{\infty}\\
&= \frac{2}{s(s^2 – 4)}\left[\frac{\pi}{2} – \frac{2}{s}\frac{\pi}{2} \right]\\
&= \frac{\pi}{s^2(s + 2)}
\end{align*}
And so
$$I(t) = \mathscr L^{-1}\left[\frac{\pi}{s^2(s + 2)}\right] = \pi\left[\frac{t}{2} + \frac{e^{-2t}}{4} – \frac{1}{4}\right]$$
Hence,
$$I(1) = \pi\left[\frac{1}{2} + \frac{e^{-2}}{4} – \frac{1}{4}
\right] = \frac{\pi}{4}\left[1 + e^{-2}\right]$$
which is correct. I'm unsure if this is mere luck or whether this is a viable method.
Has anyone used this method before?
Best Answer
Noting that the iterated integral
$$I_1=\int_0^\infty \left(\int_0^\infty e^{-st}\frac{\sin^2(tx)}{x^2(x^2+1)}\,dt\right)\,dx$$
is finite, the Fubini-Tonelli Theorem guarantees that the iterated integral formed by interchanging the order of integration,
$$I_2=\int_0^\infty \left(\int_0^\infty e^{-st}\frac{\sin^2(tx)}{x^2(x^2+1)}\,dx\right)\,dt$$
is also finite with
$$I_1=I_2$$.
But $I_2$ is the Laplace Transform of $\int_0^\infty \frac{\sin^2(tx)}{x^2(x^2+1)}\,dx$. Hence, we assert that
$$\begin{align} \mathscr{L}\left(\int_0^\infty \frac{\sin^2(tx)}{x^2(x^2+1)}\,dx\right)(s)&=\int_0^\infty \left(\int_0^\infty e^{-st}\frac{\sin^2(tx)}{x^2(x^2+1)}\,dt\right)\,dx\\\\ &=\int_0^\infty\left(\frac2s \frac{1}{(x^2+1)(4x^2+s^2)}\right)\,dx \end{align}$$
which agrees with the development in the OP!