Integral of (tg^2x+sinx)/cos^2x dx

1. Rewrite the integrand:

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

2. Integrate term-by-term:

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

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

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

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

         $$\int \frac{1}{u^{2}}\, du = - \int \frac{1}{u^{2}}\, du$$

         1. The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$:

            $$\int \frac{1}{u^{2}}\, du = - \frac{1}{u}$$

         So, the result is: $\frac{1}{u}$

      Now substitute $u$ back in:

      $$\frac{1}{\cos{\left(x \right)}}$$

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

      But the integral is

      $$- \frac{\sin{\left(x \right)}}{3 \cos{\left(x \right)}} + \frac{\sin{\left(x \right)}}{3 \cos^{3}{\left(x \right)}}$$

   The result is: $- \frac{\sin{\left(x \right)}}{3 \cos{\left(x \right)}} + \frac{\sin{\left(x \right)}}{3 \cos^{3}{\left(x \right)}} + \frac{1}{\cos{\left(x \right)}}$

3. Now simplify:

   $$\frac{\tan^{3}{\left(x \right)}}{3} + \frac{1}{\cos{\left(x \right)}}$$

4. Add the constant of integration:

   $$\frac{\tan^{3}{\left(x \right)}}{3} + \frac{1}{\cos{\left(x \right)}}+ \mathrm{constant}$$

The answer is:

$$\frac{\tan^{3}{\left(x \right)}}{3} + \frac{1}{\cos{\left(x \right)}}+ \mathrm{constant}$$