Integral of tgx-sinx dx

1. Integrate term-by-term:

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

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

      1. The integral of sine is negative cosine:

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

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

   1. Rewrite the integrand:

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

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

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

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

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

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

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

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

      Now substitute $u$ back in:

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

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

2. Add the constant of integration:

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

The answer is:

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