Integral of sinx√1-cosxdx dx

1. Integrate term-by-term:

   1. The integral of a constant times a function is the constant times the integral of the function:

      $$\int \sqrt{1} \sin{\left(x \right)}\, dx = \int \sin{\left(x \right)}\, dx$$

      1. Don't know the steps in finding this integral.

         But the integral is

         $$- \cos{\left(x \right)}$$

      So, the result is: $- \cos{\left(x \right)}$

   1. The integral of a constant times a function is the constant times the integral of the function:

      $$\int \left(- 1 \cos{\left(x \right)}\right)\, dx = - \int \cos{\left(x \right)}\, dx$$

      1. The integral of cosine is sine:

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

      So, the result is: $- \sin{\left(x \right)}$

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

2. Now simplify:

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

3. Add the constant of integration:

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

The answer is:

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