Integral of (5x^4+5sinx)dx dx

1. Integrate term-by-term:

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

      $$\int 5 x^{4}\, dx = 5 \int x^{4}\, dx$$

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

         $$\int x^{4}\, dx = \frac{x^{5}}{5}$$

      So, the result is: $x^{5}$

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

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

      1. The integral of sine is negative cosine:

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

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

   The result is: $x^{5} - 5 \cos{\left(x \right)}$

2. Add the constant of integration:

   $$x^{5} - 5 \cos{\left(x \right)}+ \mathrm{constant}$$

The answer is:

$$x^{5} - 5 \cos{\left(x \right)}+ \mathrm{constant}$$