Integral of 1/1+sinx+cosx dx

1. Integrate term-by-term:

   1. The integral of sine is negative cosine:

      $$\int \sin{\left(x \right)}\, dx = - \cos{\left(x \right)}$$

   1. The integral of cosine is sine:

      $$\int \cos{\left(x \right)}\, dx = \sin{\left(x \right)}$$

   1. The integral of a constant is the constant times the variable of integration:

      $$\int 1\, dx = x$$

   The result is: $x + \sin{\left(x \right)} - \cos{\left(x \right)}$

2. Now simplify:

   $$x - \sqrt{2} \cos{\left(x + \frac{\pi}{4} \right)}$$

3. Add the constant of integration:

   $$x - \sqrt{2} \cos{\left(x + \frac{\pi}{4} \right)}+ \mathrm{constant}$$

The answer is:

$$x - \sqrt{2} \cos{\left(x + \frac{\pi}{4} \right)}+ \mathrm{constant}$$