1. The problem is to evaluate the integral $$\int \frac{1}{\sin x + \cos x} \, dx.$$\n\n2. First, use the identity $$\sin x + \cos x = \sqrt{2} \sin \left(x + \frac{\pi}{4}\right)$$ to rewrite the integral as $$\int \frac{1}{\sqrt{2} \sin \left(x + \frac{\pi}{4}\right)} \, dx = \frac{1}{\sqrt{2}} \int \csc \left(x + \frac{\pi}{4}\right) \, dx.$$\n\n3. Let $$u = x + \frac{\pi}{4}$$ so that $$du = dx.$$ The integral becomes $$\frac{1}{\sqrt{2}} \int \csc u \, du.$$\n\n4. Recall the integral formula: $$\int \csc u \, du = -\ln \left| \csc u + \cot u \right| + C.$$\n\n5. Applying this, we get $$\frac{1}{\sqrt{2}} \left(-\ln \left| \csc u + \cot u \right| \right) + C = -\frac{1}{\sqrt{2}} \ln \left| \csc \left(x + \frac{\pi}{4}\right) + \cot \left(x + \frac{\pi}{4}\right) \right| + C.$$\n\n6. This is the final answer for the integral.