Question

Solve $\int \frac{\sin x+\cos x}{\sqrt{1+\sin 2x}}dx$, Given that x takes values for $\sin x+\cos x\ge 0$.

Answer 1

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

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

$=\int \frac{\sin x+\cos x}{\sqrt{{\sin }^{2}x+{\cos }^{2}x+2\sin x\cos x}}dx$ $\because 1=si{n}^2\theta +co{s}^2\theta$

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

$=\int \frac{\sin x+\cos x}{(\sin x+\cos x)}dx$ (Given $\sin x+\cos x\ge 0$)

$=\int 1dx$$=x$

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