Definite integral $\int_{\frac{1}{a}}^a \frac{\arctan(x)}{x}$ I have to fund the value of the above integral $$\int_{\frac{1}{a}}^a \frac{\arctan(x)}{x}$$ for $a=2014$. 
I just saw that integral of $\frac{\arctan(x)}{x}$ does not have a closed form. So I used the substitution $x=tan(\theta)$ and then used integration by parts but did not get a result. 

Let $$I(a) = \int_{\frac{1}{a}}^{a}\frac{\tan^{-1}(x)}{x}dx\;,a>0$$
Then $$I'(a) = \frac{\tan^{-1}(a)}{a}-\frac{\cot^{-1}(a)}{1}\cdot a\cdot -\frac{a}{a^2} = \frac{\tan^{-1}(a)+\cot^{-1}(a)}{a}=\frac{\pi}{2a}$$
So we get $$I(a) = \frac{\pi}{2}\ln|a|+\mathcal{C}$$
Now Put $a=1$ in $I(a)\;,$ We get $I(1) = 0$
So we get $C=0$. So $$I(a) = \frac{\pi}{2}\ln |a|$$
