Integral of 1+sin2x/sinx^2 dx

1. Integrate term-by-term:

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

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

   1. There are multiple ways to do this integral.

      Method #1

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

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

         1. Let $u = \frac{1}{\sin^{2}{\left(x \right)}}$.

            Then let $du = - \frac{2 \cos{\left(x \right)} dx}{\sin^{3}{\left(x \right)}}$ and substitute $- \frac{du}{2}$:

            $$\int \frac{1}{4 u}\, du$$

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

               $$\int \left(- \frac{1}{2 u}\right)\, du = - \frac{\int \frac{1}{u}\, du}{2}$$

               1. The integral of $\frac{1}{u}$ is $\log{\left(u \right)}$.

               So, the result is: $- \frac{\log{\left(u \right)}}{2}$

            Now substitute $u$ back in:

            $$- \frac{\log{\left(\frac{1}{\sin^{2}{\left(x \right)}} \right)}}{2}$$

         So, the result is: $- \log{\left(\frac{1}{\sin^{2}{\left(x \right)}} \right)}$

      Method #2

      1. Rewrite the integrand:

         $$\frac{\sin{\left(2 x \right)}}{\sin^{2}{\left(x \right)}} = \frac{2 \cos{\left(x \right)}}{\sin{\left(x \right)}}$$

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

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

         1. Let $u = \sin{\left(x \right)}$.

            Then let $du = \cos{\left(x \right)} dx$ and substitute $du$:

            $$\int \frac{1}{u}\, du$$

            1. The integral of $\frac{1}{u}$ is $\log{\left(u \right)}$.

            Now substitute $u$ back in:

            $$\log{\left(\sin{\left(x \right)} \right)}$$

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

   The result is: $x - \log{\left(\frac{1}{\sin^{2}{\left(x \right)}} \right)}$

2. Add the constant of integration:

   $$x - \log{\left(\frac{1}{\sin^{2}{\left(x \right)}} \right)}+ \mathrm{constant}$$

The answer is:

$$x - \log{\left(\frac{1}{\sin^{2}{\left(x \right)}} \right)}+ \mathrm{constant}$$