Ta có:
$\int\limits_0^{\frac{\pi}{4}}\dfrac{x\sin x+(x+1)\cos x}{x\sin x+c\cos x}dx$
$=\int\limits_0^{\frac{\pi}{4}}\left(1+\dfrac{x\cos x}{x\sin x+\cos x}\right)dx$
$=\int\limits_0^{\frac{\pi}{4}}dx+\int\limits_0^{\frac{\pi}{4}}\dfrac{d(x\sin x+\cos x)}{x\sin x+\cos x}$
$=\left(x+\ln(x\sin x+\cos x)\right)\left|\begin{array}{l}\dfrac{\pi}{4}\\0\end{array}\right.$
$=\dfrac{\pi}{4}+\ln\dfrac{\pi+4}{4\sqrt2}$