Неопределённый интеграл dx/(sin x+cos x)

Решите задачу:

$\displaystyle \int\limits \frac{dx}{\sin x+\cos x} = \int\limits \frac{dx}{ \sqrt{2}\sin(x+ \frac{\pi}{4} ) } =\bigg\{x+ \frac{\pi}{4}=t;\,\,\,\,\, dx=dt\bigg\}=\\ \\ \\ = \frac{1}{ \sqrt{2} } \int\limits \frac{dt}{\sin t} = \frac{1}{\sqrt{2}} \int\limits \frac{-ctgt\cdot \frac{1}{\sin t} - \frac{1}{\sin^2t} }{ctgt+ \frac{1}{\sin t} } dt=\\ \\ \\ = -\frac{1}{\sqrt{2}} \int\limits \frac{d(ctg t+ \frac{1}{\sin t})}{ctgt+ \frac{1}{\sin t} } =- \frac{1}{\sqrt{2}} \cdot \ln\bigg|ctg \, \, t+ \frac{1}{\sin t}\bigg|+C,\,\,\,\, where\,\,\,\, t=x+\frac{\pi}{4}$