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I was given the following problem:

Evaluate the following integrate using a double integral: $\int_0^{\infty}\frac{e^{-x}-e^{-2x}}{x}dx$.

The professor told us off the bat the answer was $\ln(2)$. He wants us to show our work and prove this is true. My attempt is below.

  • $I(x)$=$\int_0^{\infty}\frac{e^{-x}-e^{-2x}}{x}dx$ = $\int_0^{\infty}\frac{e^{-y}-e^{-2y}}{y}dy$ =$I(y)$.
  • Thus I can say $=I(x)=\sqrt{I(x)I(y)}$

\begin{eqnarray} \int_0^{\infty}\frac{e^{-x}-e^{-2x}}{x}dx &=&\sqrt{\int_0^{\infty}\frac{e^{-x}-e^{-2x}}{x}dx\int_0^{\infty}\frac{e^{-y}-e^{-2y}}{y}dy}\\ & = & \sqrt{\int_0^{\infty}\int_0^{\infty}\frac{(e^{-x}-e^{-2x})(e^{-y}-e^{-2y})}{xy}dxdy} \end{eqnarray}

This is where things get a little nasty. I can decided to make a change of variables. Letting $x=r\cos{\theta}$, $y=r\sin{\theta}$, thus $dA=dxdy=rdrd\theta$ by the Jacobian. Thus:

\begin{eqnarray} \int_0^{\infty}\frac{e^{-x}-e^{-2x}}{x}dx & = & \sqrt{\int_0^{2\pi}\int_0^{\infty}\frac{(e^{-r\cos{\theta}}-e^{-2r\cos{\theta}})(e^{-r\sin{\theta}}-e^{-2r\sin{\theta}})}{r^2\cos{\theta}\sin{\theta}}rdrd\theta}\\ & = & \sqrt{\int_0^{2\pi}\int_0^{\infty}\frac{(e^{-r(\cos{\theta}+\sin{\theta})}-e^{-r(2\cos{\theta}+\sin{\theta})}-e^{-r(\cos{\theta}+2\sin{\theta})}-e^{-2r(\cos{\theta}+\sin{\theta})}}{r\cos{\theta}\sin{\theta}}drd\theta} \end{eqnarray}

This is where I am stuck. From here I did a lot of trial and error trying to solve for the problem. From changing the order of integration, integration by parts, and subsitution. Is there something I am missing? Maybe a trig identity that will help simplify the expression.

Thank You for your time and I greatly appreciate any feedback you give me.

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  • $\begingroup$ $\int_{1}^2 e^{-tx} dt$ $\endgroup$
    – Yimin
    May 12 '15 at 4:17
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\begin{eqnarray} \int_0^\infty\frac{e^{-x}-e^{-2x}}{x}\,dx&=&\int_0^\infty\left[\int_1^2e^{-yx}\,dy\right]\,dx=\int_1^2\left[\int_0^\infty e^{-yx}\,dx\right]\,dy\\ &=&\int_1^2\left[-\frac{e^{-yx}}{y}\right]_0^\infty\,dy=\int_1^2\frac1y\,dy=\ln2. \end{eqnarray}

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This is called a Frullani integral. The idea is to write $$ \frac{e^{-x}-e^{-2x}}{x} = \int_1^2 e^{-tx} \, dt, $$ and then change the order of integration, which gives you $$ \int_1^2 \int_{0}^{\infty} e^{-tx} \, dx \, dt = \int_1^2 \frac{dt}{t} = \log{2}. $$

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Assume the function $\phi$ has a continuous first derivative.

Let's look at the integral

$$\int_a^b \int_c^d \varphi' (xy) dy dx$$

Note that $\frac{1}{x}\frac{\partial \varphi}{\partial y} = \frac{1}{y}\frac{\partial \varphi}{\partial x}$. Thus, we have $$\begin{align} \int_a^b \int_c^d \varphi' (xy) dy dx &=\int_a^b \int_c^d \frac{1}{x}\frac{\partial \varphi (xy)}{\partial y} dy dx \\ &=\int_a^b \frac{1}{x} \left( \varphi (dx)-\varphi (cx)\right) dx \end{align}$$

We also have

$$\begin{align} \int_a^b \int_c^d \varphi' (xy) dy dx &=\int_a^b \int_c^d \frac{1}{y}\frac{\partial \varphi (xy)}{\partial x} dy dx \\ &=\int_c^d \frac{1}{y} \left( \varphi (by)-\varphi (ay)\right) dy \end{align}$$

Assume that $\lim_{z \to \infty} \varphi (z) =0$. Then, as $a \to 0$ and $b \to \infty$, we see that

$$\begin{align} \int_0^{\infty} \int_c^d \varphi' (xy) dy dx &=- \varphi (0) \int_c^d \frac{1}{y} dx \\ &=-\varphi (0) \log (d/c) \end{align}$$

Putting it all together, we have

$$\int_0^{\infty} \frac{\left( \varphi (cx)-\varphi (dx)\right)}{x} dx =\varphi (0) \log (d/c)$$

Here, let $\phi(x)=e^{-x}$, $c=1$, and $d=2$. Then, the result is $\log2$ as expected.

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  • $\begingroup$ The prime on $\varphi$ is with respect to what? $\endgroup$
    – omegadot
    Jan 30 '16 at 13:36
  • $\begingroup$ @omegadot The convention for the prime notation means the derivative with respect to the argument. -Mark $\endgroup$
    – Mark Viola
    Jan 30 '16 at 15:24
  • $\begingroup$ So are we saying that $\frac{\partial}{\partial y} \left [\frac{1}{x} \varphi (xy) \right ] = \frac{1}{x} \frac{\partial \varphi}{\partial y} = \varphi'(xy)$ and similarly for the partial derivative with respect to $x$? $\endgroup$
    – omegadot
    Jan 31 '16 at 4:04
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Here is another approach. $$ \begin{align} \int_0^\infty\frac{e^{-x}-e^{-2x}}{x}\,\mathrm{d}x &=\int_0^\infty(e^{-x}-e^{-2x})\int_0^\infty e^{-tx}\,\mathrm{d}t\,\mathrm{d}x\\ &=\int_0^\infty\int_0^\infty(e^{-(1+t)x}-e^{-(2+t)x})\,\mathrm{d}x\,\mathrm{d}t\\ &=\int_0^\infty\left(\frac1{1+t}-\frac1{2+t}\right)\,\mathrm{d}t\\ &=\lim_{L\to\infty}\left(\int_0^L\frac1{1+t}\,\mathrm{d}t-\int_1^{L+1}\frac1{1+t}\,\mathrm{d}t\right)\\ &=\int_0^1\frac1{1+t}\,\mathrm{d}t-\lim_{L\to\infty}\int_L^{L+1}\frac1{1+t}\,\mathrm{d}t\\[9pt] &=\log(2) \end{align} $$

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