Integral of 21-cos(5x)-8x dx

1. Integrate term-by-term:

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

      $$\int \left(- 8 x\right)\, dx = - 8 \int x\, dx$$

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

         $$\int x\, dx = \frac{x^{2}}{2}$$

      So, the result is: $- 4 x^{2}$

   1. Integrate term-by-term:

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

         $$\int 21\, dx = 21 x$$

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

         $$\int \left(- \cos{\left(5 x \right)}\right)\, dx = - \int \cos{\left(5 x \right)}\, dx$$

         1. Let $u = 5 x$.

            Then let $du = 5 dx$ and substitute $\frac{du}{5}$:

            $$\int \frac{\cos{\left(u \right)}}{5}\, du$$

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

               $$\int \cos{\left(u \right)}\, du = \frac{\int \cos{\left(u \right)}\, du}{5}$$

               1. The integral of cosine is sine:

                  $$\int \cos{\left(u \right)}\, du = \sin{\left(u \right)}$$

               So, the result is: $\frac{\sin{\left(u \right)}}{5}$

            Now substitute $u$ back in:

            $$\frac{\sin{\left(5 x \right)}}{5}$$

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

      The result is: $21 x - \frac{\sin{\left(5 x \right)}}{5}$

   The result is: $- 4 x^{2} + 21 x - \frac{\sin{\left(5 x \right)}}{5}$

2. Add the constant of integration:

   $$- 4 x^{2} + 21 x - \frac{\sin{\left(5 x \right)}}{5}+ \mathrm{constant}$$

The answer is:

$$- 4 x^{2} + 21 x - \frac{\sin{\left(5 x \right)}}{5}+ \mathrm{constant}$$