Question

$\int \frac{\cos x-\sin x}{\sqrt{8-\sin 2x}}dx$

Answer 1

Putting $\cos x+\sin x=t$ in the integral

$⟹(\cos x-\sin x)dx=dt$

$(\cos x+\sin x{)}^2=t^2$

$⟹{\cos }^{2}x+{\sin }^{2}x+2\sin x\cos x=t^2$

$⟹1+\sin 2x=t^2$

$\int \frac{dx}{\sqrt{9-(1+sin2x)}}$

$\int \frac{dt}{\sqrt{9-t^2}}$

$⟹{\sin }^{-1}\frac{t}{3}+C$

$⟹{\sin }^{-1}\frac{1+\sin 2x}{3}+C$