Determine the value of the integral $$I(a)=\int_{0}^{\infty} \frac{\ln\left(a^2+x^2\right)}{b^2+x^2}dx$$
My try:
$\to I'(a)=\int_{0}^{\infty}\frac{2a}{(a^2+x^2)(b^2+x^2)}dx=\frac{\pi}{b(a+b)}$
Hence $I(a)=\frac{\pi}{b}\ln(a+b)+C$
Question: Find C???
Thank you!
Since integrand is even function, so
$\int_{0}^{\infty} \frac{\ln\left(a^2+x^2\right)}{b^2+x^2}dx$=$\frac{1}{2}\int_{-\infty}^{\infty} \frac{\ln\left(a^2+x^2\right)}{b^2+x^2}dx$
so solving the R.H.S using residues--
$\int_{C} f(z)dz=\int_{Cr} f(z)dz +\int_{-R}^{R}\frac{1}{2} \frac{\ln\left(a^2+x^2\right)}{b^2+x^2}dx$
where C is closed loop in upper half of plane and f(z)= $ \frac{1}{2}\frac{\ln\left(a^2+z^2\right)}{b^2+z^2}$
and f(z) becomes $\frac{1}{2}\frac{\ln\left(a^2+z^2\right)}{(z+bi)(z-bi)}$ by factorizing denominator
there for f(z) has singularities at z=+bi,-bi, but only z=bi lie in upper half of plane
so,$\int_{C} f(z)dz=2(pi)(i)$[residue of f(z) at $bi$]
and as R goes to infinity $\int_{Cr} f(z)dz-->0$
we get
$\int_{0}^{\infty} \frac{\ln\left(a^2+x^2\right)}{b^2+x^2}dx=\int_{C} f(z)dz=2(\pi)(i)$[residue of f(z) at $bi$]
=$\frac{\pi}{2b}\ln(a^2-b^2)$
Firstly, we introduce a lemma: \begin{equation} \int_0^{\frac{\pi}{2}}\ln(\sin(x))dx=\int_0^{\frac{\pi}{2}}\ln(\cos(x))dx\\ \mbox{Then}\\ \int_0^{\frac{\pi}{2}}\ln(\tan(x))dx=0 \end{equation} The proof see here.
Set $a=0$ and use the substitution that $x=b\tan(u)$, we have: \begin{equation} I(0)=2\int_0^\infty\frac{\ln(x)}{b^2+x^2}dx=\frac{2}{b^2}\int_0^\frac{\pi}{2}\frac{\ln(b\tan(u))}{(1+\tan(u)^2)\cos(u)^2}du\\ =\frac{2}{b^2}\int_0^\frac{\pi}{2}\ln(b\tan(u))du\\ =\frac{\pi\ln(b)}{b} \end{equation} Then, you can solve the constant $C$.
Rewrite the integral as $$ I=\int_0^{\infty} \frac{\ln \left(a^2+x^2\right)}{b^2+x^2} d x = \frac{1}{2} \int_{-\infty}^{\infty} \frac{\ln \left(a^2+x^2\right)}{b^2+x^2} d x $$ where $a,b >0$.
By the fact that $\ln(a^2+b^2)=2\Re (\ln(a+bi)),$ for any real numbers $a$ and $b$, we have $$ I=\Re\int_{-\infty}^{\infty} \frac{\ln (x+a i)}{b^2+x^2} d x . $$ Integrate $I$ using contour integration along anti-clockwise direction of the path $$\gamma=\gamma_{1} \cup \gamma_{2} \textrm{ where } \gamma_{1}(t)=t+i 0(-R \leq t \leq R) \textrm{ and } \gamma_{2}(t)=R e^{i t} (0<t<\pi) $$ yields $$ \begin{aligned} I & =\Re\left[2 \pi i \lim _{z \rightarrow b i}(z-b i) \frac{\ln (z+a i)}{x^2+b^2}\right] \\ & =\frac{\pi}{b} \Re [\ln ((b+a) i)] \\ & =\frac{\pi}{b} \ln (a+b) \end{aligned} $$
