Integral of sin(x)ln(cos(x)) dx

1. There are multiple ways to do this integral.

   Method #1

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

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

      $$\int \left(- u e^{u}\right)\, du$$

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

         $$\int u e^{u}\, du = - \int u e^{u}\, du$$

         1. Use integration by parts:

            $$\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$$

            Let $u{\left(u \right)} = u$ and let $\operatorname{dv}{\left(u \right)} = e^{u}$.

            Then $\operatorname{du}{\left(u \right)} = 1$.

            To find $v{\left(u \right)}$:

            1. The integral of the exponential function is itself.

               $$\int e^{u}\, du = e^{u}$$

            Now evaluate the sub-integral.

         2. The integral of the exponential function is itself.

            $$\int e^{u}\, du = e^{u}$$

         So, the result is: $- u e^{u} + e^{u}$

      Now substitute $u$ back in:

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

   Method #2

   1. Use integration by parts:

      $$\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$$

      Let $u{\left(x \right)} = \log{\left(\cos{\left(x \right)} \right)}$ and let $\operatorname{dv}{\left(x \right)} = \sin{\left(x \right)}$.

      Then $\operatorname{du}{\left(x \right)} = - \frac{\sin{\left(x \right)}}{\cos{\left(x \right)}}$.

      To find $v{\left(x \right)}$:

      1. The integral of sine is negative cosine:

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

      Now evaluate the sub-integral.

   2. The integral of sine is negative cosine:

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

2. Now simplify:

   $$\left(1 - \log{\left(\cos{\left(x \right)} \right)}\right) \cos{\left(x \right)}$$

3. Add the constant of integration:

   $$\left(1 - \log{\left(\cos{\left(x \right)} \right)}\right) \cos{\left(x \right)}+ \mathrm{constant}$$

The answer is:

$$\left(1 - \log{\left(\cos{\left(x \right)} \right)}\right) \cos{\left(x \right)}+ \mathrm{constant}$$