Integral of 2x(x^2+1)^9dx dx

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

   $$\int 2 x \left(x^{2} + 1\right)^{9} \cdot 1\, dx = 2 \int x \left(x^{2} + 1\right)^{9}\, dx$$

   1. There are multiple ways to do this integral.

      Method #1

      1. Let $u = x^{2} + 1$.

         Then let $du = 2 x dx$ and substitute $\frac{du}{2}$:

         $$\int \frac{u^{9}}{4}\, du$$

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

            $$\int \frac{u^{9}}{2}\, du = \frac{\int u^{9}\, du}{2}$$

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

               $$\int u^{9}\, du = \frac{u^{10}}{10}$$

            So, the result is: $\frac{u^{10}}{20}$

         Now substitute $u$ back in:

         $$\frac{\left(x^{2} + 1\right)^{10}}{20}$$

      Method #2

      1. Rewrite the integrand:

         $$x \left(x^{2} + 1\right)^{9} = x^{19} + 9 x^{17} + 36 x^{15} + 84 x^{13} + 126 x^{11} + 126 x^{9} + 84 x^{7} + 36 x^{5} + 9 x^{3} + x$$

      2. Integrate term-by-term:

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

            $$\int x^{19}\, dx = \frac{x^{20}}{20}$$

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

            $$\int 9 x^{17}\, dx = 9 \int x^{17}\, dx$$

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

               $$\int x^{17}\, dx = \frac{x^{18}}{18}$$

            So, the result is: $\frac{x^{18}}{2}$

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

            $$\int 36 x^{15}\, dx = 36 \int x^{15}\, dx$$

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

               $$\int x^{15}\, dx = \frac{x^{16}}{16}$$

            So, the result is: $\frac{9 x^{16}}{4}$

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

            $$\int 84 x^{13}\, dx = 84 \int x^{13}\, dx$$

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

               $$\int x^{13}\, dx = \frac{x^{14}}{14}$$

            So, the result is: $6 x^{14}$

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

            $$\int 126 x^{11}\, dx = 126 \int x^{11}\, dx$$

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

               $$\int x^{11}\, dx = \frac{x^{12}}{12}$$

            So, the result is: $\frac{21 x^{12}}{2}$

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

            $$\int 126 x^{9}\, dx = 126 \int x^{9}\, dx$$

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

               $$\int x^{9}\, dx = \frac{x^{10}}{10}$$

            So, the result is: $\frac{63 x^{10}}{5}$

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

            $$\int 84 x^{7}\, dx = 84 \int x^{7}\, dx$$

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

               $$\int x^{7}\, dx = \frac{x^{8}}{8}$$

            So, the result is: $\frac{21 x^{8}}{2}$

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

            $$\int 36 x^{5}\, dx = 36 \int x^{5}\, dx$$

            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: $6 x^{6}$

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

            $$\int 9 x^{3}\, dx = 9 \int x^{3}\, dx$$

            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: $\frac{9 x^{4}}{4}$

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

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

         The result is: $\frac{x^{20}}{20} + \frac{x^{18}}{2} + \frac{9 x^{16}}{4} + 6 x^{14} + \frac{21 x^{12}}{2} + \frac{63 x^{10}}{5} + \frac{21 x^{8}}{2} + 6 x^{6} + \frac{9 x^{4}}{4} + \frac{x^{2}}{2}$

   So, the result is: $\frac{\left(x^{2} + 1\right)^{10}}{10}$

2. Now simplify:

   $$\frac{\left(x^{2} + 1\right)^{10}}{10}$$

3. Add the constant of integration:

   $$\frac{\left(x^{2} + 1\right)^{10}}{10}+ \mathrm{constant}$$

The answer is:

$$\frac{\left(x^{2} + 1\right)^{10}}{10}+ \mathrm{constant}$$