Question

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

Answer 1

Now,

$\int \frac{\sin x+\cos x}{\sqrt{1+\sin 2x}}dx$

$=\int \frac{\sin x+\cos x}{\sqrt{{\sin }^{2}x+{\cos }^{2}x+2\sin x.\cos x}}dx$

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

$=\int \frac{\sin x+\cos x}{(\sin x+\cos x)}dx$

$=x+c$ [ Where $c$ is integrating constant].