(2011. A)

Lời giải

\( I = \int_0^{\frac{\pi}{4}} \frac{x \sin x + \cos x}{x \sin x + \cos x} \, dx + \int_0^{\frac{\pi}{4}} \frac{x \cos x}{x \sin x + \cos x} \, dx \)

(Vì \( (x \sin x + \cos x)' = x \cos x \))  

\( = x \Bigg|_0^{\frac{\pi}{4}} + \ln |x \sin x + \cos x| \Bigg|_0^{\frac{\pi}{4}} \)

\( = \frac{\pi}{4} + \ln \left( \frac{\sqrt{2}}{2} (1 + \frac{\pi}{4}) \right)\)