Compute $\int_{0}^{\infty}\frac{\sin(x)}{xe^x} \ dx$.

Question

I know one can solve this in many ways and the answer is $\pi/4$. However I'm interested in one particular solution involving Laplace transform.

I once saw a solution of this integral where one just did something like directly taking the Laplace transform of the integrand and then getting an integral in terms of $s$. So my question is, how does one go from

$$\mathscr{L}\left[\int_{0}^{\infty}\frac{\sin{x}}{xe^x} \ dx\right]$$

To some integral like

$$\frac{1}{2}\int_0^{\infty}\frac{1}{s^2+1} \ ds = \frac{\pi}{4}.$$

I don't remember the steps inbetween or if the first step is even correct but I wan't to know how the Laplace theory was used here. Any one who has a guess?

Answer 1

Something like this? \begin{multline} \int_0^\infty \frac{\sin x}{xe^x} dx = \int_0^\infty \sin x \int_1^\infty e^{-sx}dsdx = \int_1^\infty\!\!\int_0^\infty\sin(x) e^{-sx}dx\,ds \\= \int_1^\infty \mathcal{L}\left[\sin(x)\right](s)ds = \int_1^\infty \frac{ds}{s^2 + 1} = \frac{\pi}{2}-\tan^{-1}1 = \frac{\pi}{4} \end{multline}

Answer 2

Let $$I(a)=\int_0^\infty \frac{\sin(x)}{x}e^{-ax}dx$$ where your integral is $I(1)$. $I'(a)$ is then: $$I'(a)=-\int_0^\infty \sin(x)e^{-ax}dx$$ which is the Laplace Transform of the sine function. Thus $$I'(a)=-\frac{1}{a^2+1}$$

Integrating back we get: $$I(a)=-\arctan(a)+C$$ To find the constant, we notice that $$I(0)=\int_0^\infty \frac{\sin(x)}{x}dx=\frac{\pi}{2}$$ So we get that $C=\frac{\pi}{2}$ and that $$I(1)=-\arctan(1)+\frac{\pi}{2}=\frac{\pi}{4}$$

Answer 3

The great thing about this one is even "alternative" approaches accidentally at some point get us a double integral whose further evaluation can be seen as using a result in Laplace transforms. For example, aleden's answer begins exactly the way you would if you said "I'd rather use this", but look what it gets!

My usual preferred way of solving this sees the same phenomenon. It begins with a Schwinger parametrization, writing $\frac{1}{x}=\int_0^\infty\exp (-sx) ds$. Then the integral becomes $\int_0^\infty ds\int_0^\infty dx\sin x\exp( -(1+s)x)$. Up to a constant variable shift, this yields the deliberately Laplace-based approach in eyeballfrog's answer.

Answer 4

Here is an approach that employs the following useful property for the Laplace transform: $$\int_0^\infty f(x) g(x) \, dx = \int_0^\infty \mathcal{L} \{f(x)\} (t) \cdot \mathcal{L}^{-1} \{g(x)\} (t) \, dt.$$ Perhaps it is something like this you are looking for?

Noting that $$\mathcal{L} \{\sin x\}(t) = \frac{1}{1 + t^2},$$ and $$\mathcal{L}^{-1} \left \{\frac{e^{-x}}{x} \right \} (t)= H(t - 1).$$ Here $H(x)$ denotes the Heaviside step function. Thus

\begin{align} \int_0^\infty \frac{\sin x}{x e^x} \, dx &= \int_0^\infty \sin x \cdot \frac{e^{-x}}{x} \, dx\\ &= \int_0^\infty \mathcal{L} \{\sin x\} (t) \cdot \mathcal{L}^{-1} \left \{\frac{e^{-x}}{x} \right \} (t) \, dt\\ &= \int_0^\infty \frac{H(t - 1)}{1 + t^2} \, dt.\\ &= \int_1^\infty \frac{1}{1 + t^2} \, dt\\ &= \Big{[}\tan^{-1} t \Big{]}_1^\infty\\[1ex] &= \frac{\pi}{2} - \frac{\pi}{4}\\[1ex] &= \frac{\pi}{4}. \end{align}