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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