Integral of 2sin(x)cos(5x)cos(x)dx dx

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

   $$\int 2 \sin{\left(x \right)} \cos{\left(5 x \right)} \cos{\left(x \right)} 1\, dx = 2 \int \sin{\left(x \right)} \cos{\left(x \right)} \cos{\left(5 x \right)}\, dx$$

   1. Rewrite the integrand:

      $$\sin{\left(x \right)} \cos{\left(x \right)} \cos{\left(5 x \right)} = 16 \sin{\left(x \right)} \cos^{6}{\left(x \right)} - 20 \sin{\left(x \right)} \cos^{4}{\left(x \right)} + 5 \sin{\left(x \right)} \cos^{2}{\left(x \right)}$$

   2. Integrate term-by-term:

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

         $$\int 16 \sin{\left(x \right)} \cos^{6}{\left(x \right)}\, dx = 16 \int \sin{\left(x \right)} \cos^{6}{\left(x \right)}\, dx$$

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

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

            $$\int u^{6}\, du$$

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

               $$\int \left(- u^{6}\right)\, du = - \int u^{6}\, du$$

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

                  $$\int u^{6}\, du = \frac{u^{7}}{7}$$

               So, the result is: $- \frac{u^{7}}{7}$

            Now substitute $u$ back in:

            $$- \frac{\cos^{7}{\left(x \right)}}{7}$$

         So, the result is: $- \frac{16 \cos^{7}{\left(x \right)}}{7}$

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

         $$\int \left(- 20 \sin{\left(x \right)} \cos^{4}{\left(x \right)}\right)\, dx = - 20 \int \sin{\left(x \right)} \cos^{4}{\left(x \right)}\, dx$$

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

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

            $$\int u^{4}\, du$$

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

               $$\int \left(- u^{4}\right)\, du = - \int u^{4}\, du$$

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

                  $$\int u^{4}\, du = \frac{u^{5}}{5}$$

               So, the result is: $- \frac{u^{5}}{5}$

            Now substitute $u$ back in:

            $$- \frac{\cos^{5}{\left(x \right)}}{5}$$

         So, the result is: $4 \cos^{5}{\left(x \right)}$

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

         $$\int 5 \sin{\left(x \right)} \cos^{2}{\left(x \right)}\, dx = 5 \int \sin{\left(x \right)} \cos^{2}{\left(x \right)}\, dx$$

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

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

            $$\int u^{2}\, du$$

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

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

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

                  $$\int u^{2}\, du = \frac{u^{3}}{3}$$

               So, the result is: $- \frac{u^{3}}{3}$

            Now substitute $u$ back in:

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

         So, the result is: $- \frac{5 \cos^{3}{\left(x \right)}}{3}$

      The result is: $- \frac{16 \cos^{7}{\left(x \right)}}{7} + 4 \cos^{5}{\left(x \right)} - \frac{5 \cos^{3}{\left(x \right)}}{3}$

   So, the result is: $- \frac{32 \cos^{7}{\left(x \right)}}{7} + 8 \cos^{5}{\left(x \right)} - \frac{10 \cos^{3}{\left(x \right)}}{3}$

2. Now simplify:

   $$\frac{2 \left(- 48 \cos^{4}{\left(x \right)} + 84 \cos^{2}{\left(x \right)} - 35\right) \cos^{3}{\left(x \right)}}{21}$$

3. Add the constant of integration:

   $$\frac{2 \left(- 48 \cos^{4}{\left(x \right)} + 84 \cos^{2}{\left(x \right)} - 35\right) \cos^{3}{\left(x \right)}}{21}+ \mathrm{constant}$$

The answer is:

$$\frac{2 \left(- 48 \cos^{4}{\left(x \right)} + 84 \cos^{2}{\left(x \right)} - 35\right) \cos^{3}{\left(x \right)}}{21}+ \mathrm{constant}$$