Integral of (-84+18x^5+3Sinx)dx dx

1. Integrate term-by-term:

   1. Integrate term-by-term:

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

         $$\int 18 x^{5}\, dx = 18 \int x^{5}\, dx$$

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

            $$\int x^{5}\, dx = \frac{x^{6}}{6}$$

         So, the result is: $3 x^{6}$

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

         $$\int \left(-84\right)\, dx = - 84 x$$

      The result is: $3 x^{6} - 84 x$

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

      $$\int 3 \sin{\left(x \right)}\, dx = 3 \int \sin{\left(x \right)}\, dx$$

      1. The integral of sine is negative cosine:

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

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

   The result is: $3 x^{6} - 84 x - 3 \cos{\left(x \right)}$

2. Add the constant of integration:

   $$3 x^{6} - 84 x - 3 \cos{\left(x \right)}+ \mathrm{constant}$$

The answer is:

$$3 x^{6} - 84 x - 3 \cos{\left(x \right)}+ \mathrm{constant}$$