|
|
$I=\int\limits_{0}^{\frac{\pi}{4} }\frac{\sin x+\cos x}{3+\sin 2x} dx =\frac{1}{4}\int\limits_{0}^{\frac{\pi}{4} }(\sin x+\cos x)\frac{2-\sin x+\cos x+2+\sin x-\cos x}{4-(\sin x- \cos x)^2} dx$
$=\frac{1}{4}\int\limits_{0}^{\frac{\pi}{4} }\frac{\sin x+\cos x}{\sin x-\cos x+2} dx+\frac{1}{4}\int\limits_{0}^{\frac{\pi}{4} }\frac{\sin x+\cos x}{-\sin x+\cos x+2} dx$
$=\frac{1}{4}\ln\left| {\sin x-\cos x+2} \right|_{0}^{\frac{\pi}{4} }-\frac{1}{4}\ln\left| {-\sin x+\cos x+2} \right|_{0}^{\frac{\pi}{4} }$
$I=\frac{1}{4}\ln3$
|