Question

Solve: $\int \frac{\cos x-\sin x}{\sqrt{8+\sin 2x}}$

Answer 1

$\int \frac{\cos x-\sin x}{\sqrt{8+\sin 2x}}=\int \frac{\cos x-\sin x}{\sqrt{7+1+\sin 2x}}=\int \frac{\cos x-\sin x}{\sqrt{7+{\sin }^{2}x+{\cos }^{2}x+2\sin x\cos x}}=\int \frac{\cos x-\sin x}{\sqrt{7+{(\sin x+\cos x)}^2}}\sin x+\cos x=tdx(-\sin x+\cos x)=dtdx(\cos x-\sin x)=dt\int \frac{dx}{\sqrt{7+t^2}}=\int \frac{dx}{\sqrt{{\left(\sqrt{7}\right)}^2+t^2}}$