Question

$\int_{\pi/4}^{\pi/3} \frac{\ln(tanx)}{\sin x \cos x} dx$

Answer 1

$\frac{d}{dx}\left( \ln \left( \tan x\right) \right) ^{2}=\frac{2}{\tan x} \frac{1}{\cos ^{2}x}\left( \ln \left( \tan x\right) \right) =\frac{2\ln \left( \tan x\right) }{\sin x\cos x}$
olduğundan
$$\int\limits_{\frac{\pi }{4}}^{\frac{\pi }{3}}\frac{\ln \left( \tan x\right) }{\sin x\cos x}dx=\left. \frac{1}{2}\left( \ln \left( \tan x\right) \right) ^{2}\right\vert _{\frac{\pi }{4}}^{\frac{\pi }{3}}=\frac{1}{2}\left( \left( \ln \sqrt{3}\right) ^{2}-\left( \ln 1\right) ^{2}\right) =\frac{1}{8}\ln ^{2}3$$
dir.