A problem proposed by prof. Gadjev

Evaluate 

$$\int_0^{\infty}\frac{\ln x}{x^2+2x+5}\, \text{d}x$$

$\textbf{Solution.}$

The indefinite integral is not expressible in elementary functions. Denote the integral by $I$. First make the change of variables $\displaystyle x\to \frac{1}{x}$ to obtain 

$$I=-\int_0^{\infty}\frac{\ln x}{5x^2+2x+1}\, \text{d}x\ \ $$

In the original integral make the change of variables $x\to 5x$ to obtain 

$$ I=\int_0^{\infty}\frac{\ln (5x)}{25x^2+10x+5}5\, \text{d}x=\int_0^{\infty}\frac{\ln x}{5x^2+2x+1}\, \text{d}x+\int_0^{\infty}\frac{\ln 5}{5x^2+2x+1}\, \text{d}x=$$

$$-I+\int_0^{\infty}\frac{\ln 5}{5x^2+2x+1}\, \text{d}x$$

whence 

$$I=\frac{1}{2} \int_0^{\infty}\frac{\ln 5}{5x^2+2x+1}\, \text{d}x=\frac{1}{8}\arctan(2)\ln(5)$$

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