Integral of 7x-2x³÷1+7x⁴ dx

1. Integrate term-by-term:

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

      $$\int 7 x^{4}\, dx = 7 \int x^{4}\, dx$$

      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: $\frac{7 x^{5}}{5}$

   1. Integrate term-by-term:

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

         $$\int 7 x\, dx = 7 \int x\, dx$$

         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: $\frac{7 x^{2}}{2}$

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

         $$\int \left(- \frac{2 x^{3}}{1}\right)\, dx = - \int 2 x^{3}\, dx$$

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

            $$\int 2 x^{3}\, dx = 2 \int x^{3}\, dx$$

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

               $$\int x^{3}\, dx = \frac{x^{4}}{4}$$

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

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

      The result is: $- \frac{x^{4}}{2} + \frac{7 x^{2}}{2}$

   The result is: $\frac{7 x^{5}}{5} - \frac{x^{4}}{2} + \frac{7 x^{2}}{2}$

2. Now simplify:

   $$\frac{x^{2} \left(14 x^{3} - 5 x^{2} + 35\right)}{10}$$

3. Add the constant of integration:

   $$\frac{x^{2} \left(14 x^{3} - 5 x^{2} + 35\right)}{10}+ \mathrm{constant}$$

The answer is: