Question

integral sin x + cos x upon under root 1 + sin 2x DX​

Answer 1

Answer:

$\large\bold\red{x+c}$

Step-by-step Explanation:

We have to integrate,

$\displaystyle \int \frac{ \sin(x) + \cos(x) }{ \sqrt{1 + \sin(2x) } } dx \\\\ \\ = \displaystyle \int \frac{ \sin(x) + \cos(x) }{ \sqrt{1 + 2 \sin(x) \cos(x) } } dx \\ \\\\ = \displaystyle \int \frac{ \sin(x) + \cos(x) }{ \sqrt{ { \sin}^{2} x + { \cos}^{2}x + 2 \sin(x) \cos(x) }} dx \\ \\\\ = \displaystyle \int \frac{ \sin(x) + \cos(x) }{ \sqrt{ {( \sin x + \cos x) }^{2} } } dx \\\\ \\ = \displaystyle \int \frac{ \sin(x) + \cos(x) }{ \sin(x) + \cos(x) } dx \\ \\\\ = \displaystyle \int dx \\ \\\\ = x + c$

where, c is any integral constant.