Nguyên hàm của hàm số \(f(x) ={\cos 2x\ln \left( {\sin x + \cos x} \right)dx} \)  là:

Ta có:

\(\begin{array}{l}\cos 2x\ln \left( {\sin x + \cos x} \right) = \left( {\cos x + \sin x} \right)\left( {\cos x - \sin x} \right)\ln \left( {\sin x + \cos x} \right)\\ \Rightarrow I = \int {\left( {\cos x + \sin x} \right)\left( {\cos x - \sin x} \right)\ln \left( {\sin x + \cos x} \right)dx} \end{array}\)

Đặt \(t = \sin x + \cos x \Rightarrow dt = \left( {\cos x - \sin x} \right)dx\) , khi đó ta có:\(I = \int {t\ln tdt} \)

Đặt $\left\{ \begin{array}{l}u = \ln t\\dv = tdt\end{array} \right. \Rightarrow \left\{ \begin{array}{l}du = \dfrac{1}{t}dt\\v = \dfrac{{{t^2}}}{2}\end{array} \right.$

$\begin{array}{l} \Rightarrow I = \dfrac{1}{2}{t^2}\ln t - \dfrac{1}{2}\int {tdt}  + C = \dfrac{1}{2}{t^2}\ln t - \dfrac{{{t^2}}}{4} + {C_1}\\\,\,\,\,\,\,\,\,\, = \dfrac{1}{2}{\left( {\sin x + \cos x} \right)^2}\ln \left( {\sin x + \cos x} \right) - \dfrac{{{{\left( {\sin x + \cos x} \right)}^2}}}{4} + {C_1}\\\,\,\,\,\,\,\,\,\, = \dfrac{1}{2}\left( {{{\sin }^2}x + {{\cos }^2}x + \sin 2x} \right)\ln \left( {\sin x + \cos x} \right) - \dfrac{{1 + \sin 2x}}{4} + {C_1}\\\,\,\,\,\,\,\,\,\, = \dfrac{1}{4}\left( {1 + \sin 2x} \right)\ln {\left( {\sin x + \cos x} \right)^2} - \dfrac{{\sin 2x}}{4} - \dfrac{1}{4} + {C_1}\\\,\,\,\,\,\,\,\,\, = \dfrac{1}{4}\left( {1 + \sin 2x} \right)\ln \left( {1 + \sin 2x} \right) - \dfrac{{\sin 2x}}{4} + C.\end{array}$