Đáp án:

• \( \int \frac{\sin x (2 \cos x + 1)}{\sqrt{1 + 3 \cos x}} \, dx \)

• Đặt \( t = \sqrt{1 + 3 \cos x} \Rightarrow \cos x = \frac{t^2 - 1}{3} \Rightarrow -\sin x \, dx = \frac{2}{3} t \, dt \)

• \( \int \frac{\sin x (2 \cos x + 1)}{\sqrt{1 + 3 \cos x}} \, dx = \int -\frac{2}{3} t \left( \frac{2(t^2 - 1)}{3}  + 1 \right) \frac{1}{t} \, dt \)

\( = -\frac{2}{3} \int \frac{(2t^2 + 1)}{3} \, dt = -\frac{2}{9} \int (2t^2 + 1) \, dt \)

\( = -\frac{2}{9} \left[ \frac{2t^3}{3} + t \right] + C \)

\( = -\frac{4}{27} t^3 - \frac{2}{9} t + C \)