Cho hàm số $f\left( x \right)$ liên tục và có đạo hàm trên $\left(...

Ta có:


Đặt: $\left\{ \begin{aligned}
& u=x \\
& \text{d}v=\dfrac{1}{{{\cos }^{2}}x} \\
\end{aligned} \right.\Rightarrow \left\{ \begin{aligned}
& \text{d}u=\text{d}x \\
& v=\tan x \\
\end{aligned} \right.$$\Rightarrow \int{\dfrac{x}{{{\cos }^{2}}x}}\text{d}x=x\tan x-\int{\tan x\text{d}x}=x\tan x+\ln (\cos x)+Cf\left( x \right)\sin x=x\tan x+\ln (\cos x)+Cx=\dfrac{\pi }{3}\dfrac{\sqrt{3}}{2}f\left( \dfrac{\pi }{3} \right)=\dfrac{\sqrt{3}\pi }{3}+\ln \dfrac{1}{2}+Cx=\dfrac{\pi }{6}\dfrac{1}{2}f\left( \dfrac{\pi }{6} \right)=\dfrac{\sqrt{3}\pi }{18}+\ln \dfrac{\sqrt{3}}{2}+C\sqrt{3}f\left( \dfrac{\pi }{3} \right)-f\left( \dfrac{\pi }{6} \right)=\dfrac{5\pi \sqrt{3}}{9}-\ln 3\Rightarrow a=\dfrac{5}{9} ; b=-1\Rightarrow a+b=-\dfrac{4}{9}$