Problema Evaluar la siguiente integral $$\int_0^{2*\pi}\sqrt{1+\sin x}dx$ $
Intento de solución
Tenga en cuenta que\begin{align*} & \int_{0}^{2\pi}\sqrt{1+\sin x}dx\\ = & \int_{0}^{\pi}\sqrt{1+\sin x}dx+\int_{\pi}^{2\pi}\sqrt{1+\sin x}dx\\ = & \int_{0}^{\pi}\sqrt{\left(\cos\frac{x}{2}+\sin\frac{x}{2}\right)^{2}}dx+\int_{0}^{\pi}\sqrt{1+\sin\left(x-\pi\right)}d\left(x-\pi\right)\\ = & \int_{0}^{\pi}\sqrt{\left(\cos\frac{x}{2}+\sin\frac{x}{2}\right)^{2}}dx+\int_{0}^{\pi}\sqrt{1-\sin x}dx\\ = & \int_{0}^{\pi}\sqrt{\left(\cos\frac{x}{2}+\sin\frac{x}{2}\right)^{2}}dx+\int_{0}^{\pi}\sqrt{\left(\cos\frac{x}{2}-\sin\frac{x}{2}\right)^{2}}dx\\ = & \int_{0}^{\pi}\left|\cos\frac{x}{2}+\sin\frac{x}{2}\right|dx+\int_{0}^{\pi}\left|\cos\frac{x}{2}-\sin\frac{x}{2}\right|dx\\ = & \int_{0}^{\frac{\pi}{2}}\cos\frac{x}{2}+\sin\frac{x}{2}dx+\int_{\frac{\pi}{2}}^{\pi}\cos\frac{x}{2}+\sin\frac{x}{2}+\int_{0}^{\frac{\pi}{2}}\cos\frac{x}{2}-\sin\frac{x}{2}dx+\int_{\frac{\pi}{2}}^{\pi}\sin\frac{x}{2}-\cos\frac{x}{2}dx\\ = & \int_{0}^{\frac{\pi}{2}}2\cos\frac{x}{2}dx+\int_{\frac{\pi}{2}}^{\pi}\sin\frac{x}{2}dx\\ = & \left.4\sin\frac{x}{2}\right|_{0}^{\frac{\pi}{2}}-\left.4\cos\frac{x}{2}\right|_{\frac{\pi}{2}}^{\pi}=\boxed{4\sqrt{2}}. \end{align*}
Pregunta No sé si resolví esta integral de la mejor manera posible. Realmente agradezco si alguien puede ofrecer alguna solución alternativa.
