Computing an integral?

Question

Is that true $\int_0^{+\infty} \cfrac{\ln x}{a^2+x^2} dx = \cfrac{\pi\ln a}{2a}$ , where $a>0$?

And how to compute?

Answer 1

Putting $x=a\tan t,dx=a\sec^2tdt$ and $x=0\implies t=0, x=\infty\implies t=\frac\pi2$

$$\int_0^{+\infty} \cfrac{\ln x}{a^2+x^2} \, dx=\frac1a\int_0^{\frac\pi2}\ln(a\tan t)\,dt$$

$$\int_0^{\frac\pi2}\ln(a\tan t)dt=\ln a\int_0^{\frac\pi2}\, dt+\int_0^{\frac\pi2}\ln \tan t \, dt$$

Now,

$$\int_0^{\frac\pi2}\ln \tan t \,dt$$ $$=\int_0^{\frac\pi2}\ln \tan (\frac\pi2+0-t) \,dt$$

(as $\int_a^bf(a+b-x)dx=\int_b^af(t)(-dt)$ (putting $a+b-x=t$)

$\int_a^bf(a+b-x)dx=\int_a^bf(t)dt=\int_a^bf(x)dx$)

$$=\int_0^{\frac\pi2}\ln \cot t dt$$ $$=-\int_0^{\frac\pi2}\ln \tan t\,dt\implies \int_0^{\frac\pi2}\ln \tan t\, dt=0$$

and

$$\int_0^{\frac\pi2} \,dt=t\mid _0^{\frac\pi2}=\frac\pi2$$

Answer 2

This integral can be easiest integrated using the method explained here.

Define the function $$f(z) = \frac{\ln^2 z}{a^2+z^2}.$$ With the branch cut of the logarithm along the negative real axis, we choose the keyhole contour

It is easy to check that the function falls of at $\infty$ sufficiently fast that there is no contribution from the large circle. The integration along the branch cut yields $$4\pi i \int_0^\infty \frac{\ln x}{a^2+x^2} dx$$ which has to be the same as the contribution from the residue at $x=\pm ia$, $$2\pi i [ \mathop{\rm Res}_{x=a} f(z) + \mathop{\rm Res}_{x=-a} f(z) ] = \frac\pi{a} \underbrace{[\ln^2(ia) - \ln^2(-ia)]}_{=2i\pi\ln a}.$$

So we have $$\int_0^\infty \frac{\ln x}{a^2+x^2} dx = \frac{\pi \ln a}{2 a}.$$

Answer 3

Take the following complex contour:

$$C_R:=[-R,R]\cup\gamma_R:=\{z\in\Bbb C\;;\;z=Re^{it}\,\,,\,0\leq t\leq \pi\}$$

Within the above path the function has one simple pole, $\,z= ai\,$ (note that $\,z=0\, $ is not a pole. Why?), and

$$Res_{z=ai}(f)=\lim_{z\to ai}\frac{(z-i\sqrt a)Log(z)}{a^2+z^2}=\frac{Log(ai)}{2ai}=\frac{\log a+\frac{\pi i}{2}}{2ai}$$

when we denote $\,Log (z)=$ the complex logarithm, and $\,\log z=$ the real one.

We also have, by Cauchy's estimation formula, that

$$\left|\int_{\gamma_R}\frac{Log(z)}{z^2+a^2}dz\right|\leq \max_{z\in\gamma_R}\frac{|Log(z)|}{|z^2+a^2|}\cdot\pi R\leq \frac{\sqrt 2\,\pi R\log R}{R^2-a^2}\xrightarrow [R\to\infty]{} 0$$

since $|Log (z)|=|\log z+i\arg z|=\sqrt{\log^2R+\arg^2(z)}\leq\sqrt{\log^2R+\pi^2}\leq\sqrt 2\,\log R\,$

for $\,R\,$ big enough.

Thus, using Cauchy's Residue theorem and then passing to the limit:

$$2\pi i\frac{\log a+\frac{\pi i}{2}}{2ai}=\oint_{C_R}f(z)\,dz=\int_{-R}^R\frac{\log |x|}{x^2+a^2}dx+\int_{\gamma_R}f(z)\,dz\xrightarrow[R\to\infty]{} \int_{-\infty}^\infty\frac{\log |x|}{x^2+a^2}dx$$

Since the last integral's integrand function is even, and comparing real and imaginary parts, we get:

$$\int_0^\infty\frac{\log x}{x^2+a^2}dx=\frac{\pi\,\log a}{2a}$$

Answer 4

Consider the parameter-dependent integral:

$$I(a,b)=\int_0^{\infty}\frac{\ln(b^2+x^2)}{a^2+x^2}dx$$

Let's differentiate with respect to $b$:

$$\frac{\partial I(a,b) }{\partial b}=\int_0^{\infty}\frac{2b}{(b^2+x^2)(a^2+x^2)}dx=\frac{\pi}{a(a+b)}$$

Now, let's integrate the last with respect to $b$:

$$I(a,b)=\pi\frac{\ln(a+b)}{a}+C$$

$C=0$ because $I(\infty,b)=0$

And the original integral:

$$I=\frac{1}{2}I(a,0)=\frac{\pi}{2}\frac{\ln a}{a}$$