How to find constant for feynman's technique of integration $\int_{0}^{\infty}\frac{\ln\left(x^{2}+1\right)}{x^{2}+1}dx$

Question

I've got an integral $$\int_{0}^{\infty}\frac{\ln\left(x^{2}+1\right)}{x^{2}+1}dx$$ and when I used Feynman's technique of integration $$I(t) = \int_{0}^{\infty}\frac{\ln\left(x^{2}+t\right)}{x^{2}+1}dx$$ I got the result $$\pi\cdot\ln\left(\sqrt{t}+1\right)$$ but the problem is that after integrating $I(t)$, I must include constant $C$ so the final result should be $$I(t)=\pi\cdot\ln\left(\sqrt{t}+1\right) + C$$ and I have no idea how to calculate that $C=0$

Answer 1

$$I(0) = 2\int_{0}^{\infty}\frac{\ln\left(x\right)}{x^{2}+1}dx$$

Let $t=1/x$

$$I(0) = -2\int_{0}^{\infty}\frac{\ln\left(t\right)}{t^{2}+1}dt$$

Add them

$$I(0)=0~~\Longrightarrow ~~C=0$$