Definite integral $\arctan$ over $x$

Asked Oct 17 '21 at 05:39

Active Oct 17 '21 at 05:49

Viewed 27 times

$$\int_0^{\infty}\frac{\arctan(2x)-\arctan(x)}{x }dx$$

I checked that the integral converges. Next, the only thing that seems to appear to me to do is arctan sum formula $$\int_0^{\infty}\frac{\arctan({x\over{1+2x^2}})}{x }dx$$ but then it's dead end. How do I proceed further?

edited Oct 17 '21 at 05:49

Gary

31,845

asked Oct 17 '21 at 05:39

RKK

1

https://en.wikipedia.org/wiki/Frullani_integral – Gary Oct 17 '21 at 05:48
The answer to the question Solution of $\int_0^{\infty} \frac{\arctan(x)-\arctan(2x)}x\mathrm dx$ : Why Wolfram Alpha shows a different answer than mine? contains a computation that will lead you to the answer. In addition, consider applying Frullani's theorem. – Sarvesh Ravichandran Iyer Oct 17 '21 at 05:58

Definite integral $\arctan$ over $x$

0 Answers0