Question

Write a value of $\int \frac{\sin x+\cos x}{\sqrt{1+\sin 2x}}dx$.

Answer 1

$$\begin{gathered} \int \left(\frac{\sin x+\cos x}{\sqrt{1+\sin 2x}}\right)dx \\ \Rightarrow \int \frac{\left(\sin x+\cos x\right)dx}{\sqrt{{\sin }^{2}x+{\cos }^{2}x+2\sin x\cos x}} \\ \Rightarrow \int \frac{\left(\sin x+\cos x\right)dx}{\sqrt{{\left(\sin x+\cos x\right)}^2}} \\ \Rightarrow \int dx \\ \Rightarrow x+C \end{gathered}$$