$\displaystyle \int_{\frac{3\pi}{4}}^{\frac{7\pi}{4}} \dfrac{\sin x + \cos x}{\sqrt{1 + \sin 2x}} \, dx$ is equal to

Given integral: $\int \dfrac{\sin x + \cos x}{\sqrt{1 + \sin 2x}} \, dx$. We know $\sin 2x = 2\sin x \cos x$ and $1 + \sin 2x = (\sin x + \cos x)^2$. So, $\sqrt{1 + \sin 2x} = |\sin x + \cos x|$. Hence, integrand becomes $\dfrac{\sin x + \cos x}{|\sin x + \cos x|} = 1$. Therefore, $\int dx = x + C$. $\boxed{\text{Answer: (B) } x}$