|
|
$I=\int\limits_{0}^{\frac{\pi}{4} }\frac{(\sin x+2\cos x)}{3\sin x+\cos
x} dx=\frac{1}{2}\int\limits_{0}^{\frac{\pi}{4} }(1+\frac{3\cos x-\sin
x}{3\sin x+\cos x} )dx$$=\frac{1}{2}\left[\int\limits_{0}^{\frac{\pi}{4}
}dx+\int\limits_{0}^{\frac{\pi}{4} }\frac{d(3\sin x+\cos x)}{3\sin
x+\cos x} \right]$
$I=\frac{1}{2}[x+\ln (3\sin x+\cos x)]|_{0}^{\frac{\pi}{4} }=\frac{1}{2}[(\frac{\pi}{4}+\ln 2 {\sqrt{2}} )-(0+\ln 1)] $
$I=\frac{1}{8}(\pi+6\ln 2) $
|