|
|
a)
$\dfrac{1}{\tan x+1}=\dfrac{\cos x}{\sin x +\cos x}=\dfrac{1}{2}+\dfrac{1}{2}.\dfrac{\cos x-\sin x}{\sin x +\cos x}=\dfrac{1}{2}(x)'+ \dfrac{1}{2}(\ln |\sin x+ \cos x|)'$
suy ra
$\int\limits_{0}^{\pi/4}\dfrac{1}{\tan x+1}dx=\dfrac{1}{2}\left[ {x+\ln |\sin x+ \cos x|} \right]_{0}^{\pi/4}=\dfrac{\pi+\ln 4}{8}$
|