Integral of sin2x(secx)dx dx

1. Rewrite the integrand:

   $$\sin{\left(2 x \right)} \sec{\left(x \right)} 1 = 2 \sin{\left(x \right)}$$

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

   $$\int 2 \sin{\left(x \right)}\, dx = 2 \int \sin{\left(x \right)}\, dx$$

   1. The integral of sine is negative cosine:

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

   So, the result is: $- 2 \cos{\left(x \right)}$

3. Add the constant of integration:

   $$- 2 \cos{\left(x \right)}+ \mathrm{constant}$$

The answer is:

$$- 2 \cos{\left(x \right)}+ \mathrm{constant}$$