Tích phân \(I=\int_{0}^{2 \pi} \sqrt{1+\sin x} d x...

A

- Hướng dẫn giải

\(\begin{aligned} I &=\int_{0}^{2 \pi} \sqrt{\left(\sin \frac{x}{2}+\cos \frac{x}{2}\right)^{2}} d x=\int_{0}^{2 \pi}\left|\sin \frac{x}{2}+\cos \frac{x}{2}\right| d x=\sqrt{2} \int_{0}^{2 \pi}\left|\sin \left(\frac{x}{2}+\frac{\pi}{4}\right)\right| d x \\ &=\sqrt{2}\left[\int_{0}^{\frac{3 \pi}{2}} \sin \left(\frac{x}{2}+\frac{\pi}{4}\right) d x-\int_{\frac{3 \pi}{2}}^{2 \pi} \sin \left(\frac{x}{2}+\frac{\pi}{4}\right) d x\right]=4 \sqrt{2} \end{aligned}\)