Integral of (3sinx-5x+1)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:

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

         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: $- \frac{5 x^{2}}{2}$

      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: $- \frac{5 x^{2}}{2} - 3 \cos{\left(x \right)}$

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

      $$\int 1\, dx = x$$

   The result is: $- \frac{5 x^{2}}{2} + x - 3 \cos{\left(x \right)}$

2. Add the constant of integration:

   $$- \frac{5 x^{2}}{2} + x - 3 \cos{\left(x \right)}+ \mathrm{constant}$$

The answer is:

$$- \frac{5 x^{2}}{2} + x - 3 \cos{\left(x \right)}+ \mathrm{constant}$$