Question

$\displaystyle \sf \int_{0}^{ \infty} \frac{ \arctan(x) - \arctan(2x) + \arctan(3x) - \arctan(4x)}{x} \: dx$​

Answer 1

Answer:

Because

arctan(−2)+arctan(−3)=−135∘<0

By the way, tan(−135∘)=1.

please make as Brainleast

Answer 2

$\tiny \displaystyle \sf \int_{0}^{ \infty} \frac{ \arctan(x) - \arctan(2x) + \arctan(3x) - \arctan(4x)}{x} \: dx$

$\tiny \displaystyle \sf \int_{0}^{ \infty} \frac{ \arctan(x) - \arctan(2x)}{x} \: dx + \int_{0}^{ \infty} \frac{ \arctan(3x) - \arctan(4x) }{x} \: dx$

$\sf \frac{\pi}{2} \ln \frac{1}{2} + \frac{\pi}{2} \ln \frac{3}{4}$

$\frac{\pi}{2} \ln \frac{3}{8}$