Integral of (1+0,7*x)^2 dx

1. There are multiple ways to do this integral.

   Method #1

   1. Let $u = \frac{7 x}{10} + 1$.

      Then let $du = \frac{7 dx}{10}$ and substitute $\frac{10 du}{7}$:

      $$\int \frac{10 u^{2}}{7}\, du$$

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

         $$\int u^{2}\, du = \frac{10 \int u^{2}\, du}{7}$$

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

            $$\int u^{2}\, du = \frac{u^{3}}{3}$$

         So, the result is: $\frac{10 u^{3}}{21}$

      Now substitute $u$ back in:

      $$\frac{10 \left(\frac{7 x}{10} + 1\right)^{3}}{21}$$

   Method #2

   1. Rewrite the integrand:

      $$\left(\frac{7 x}{10} + 1\right)^{2} = \frac{49 x^{2}}{100} + \frac{7 x}{5} + 1$$

   2. Integrate term-by-term:

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

         $$\int \frac{49 x^{2}}{100}\, dx = \frac{49 \int x^{2}\, dx}{100}$$

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

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

         So, the result is: $\frac{49 x^{3}}{300}$

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

         $$\int \frac{7 x}{5}\, dx = \frac{7 \int x\, dx}{5}$$

         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: $\frac{7 x^{2}}{10}$

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

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

      The result is: $\frac{49 x^{3}}{300} + \frac{7 x^{2}}{10} + x$

2. Now simplify:

   $$\frac{\left(7 x + 10\right)^{3}}{2100}$$

3. Add the constant of integration:

   $$\frac{\left(7 x + 10\right)^{3}}{2100}+ \mathrm{constant}$$

The answer is:

$$\frac{\left(7 x + 10\right)^{3}}{2100}+ \mathrm{constant}$$