Question

The value of $\int \frac{\sin \left(x\right)+\cos \left(x\right)}{\sqrt{1+2\sin \left(x\right)}}·dx$ is

• A. $\sqrt{(1+\cos (2x\left)\right)}+c$
• B. $\sqrt{x}-c$
• C. $x+c$
• D. $\sqrt{x+2x}+c$

Answer 1

The correct option is C

$x+c$

Explanation for the correct option:

Compute the required value:

$$\begin{gathered} \int \frac{\sin \left(x\right)+\cos \left(x\right)}{\sqrt{1+\sin \left(2x\right)}}·dx \\ {\left(\sin \left(x\right)+\cos \left(x\right)\right)}^2={\sin }^2\left(x\right)+{\cos }^2\left(x\right)+2·\sin \left(x\right)·\cos \left(x\right) \\ \Rightarrow {\left(\sin \left(x\right)+\cos \left(x\right)\right)}^2=1+\sin \left(2x\right);\left[2·\sin \left(x\right)·\cos \left(x\right)=\sin \left(2x\right)\right] \end{gathered}$$

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

where $c$ is the constant

Hence, option $C$ is the correct answer.