Evaluate $$\int \frac{dx}{\sin{x}-\cos{x}} $$
I know it can be done by Weierstrass substitution. But I am looking for new/simple approach. For example I tried:
$$\int \frac{1}{\sin{x}-\cos{x}} \cdot \frac{\sin{x}+\cos{x}}{\sin{x}+\cos{x}}dx=\int \frac{\sin{x}+\cos{x}}{-\cos{2x}}dx ,$$
but I can't continue from here.
Best Answer
Write $$\sin{x}-\cos{x}=\sqrt2\left(\frac1{\sqrt2}\sin{x}-\frac1{\sqrt2}\cos{x}\right)=\sqrt2\sin\left(x-\frac{\pi}4\right).$$