Integral of 2x(x^2+1)^5 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 $du$:

      $$\int u^{5}\, du$$

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

         $$\int u^{5}\, du = \frac{u^{6}}{6}$$

      Now substitute $u$ back in:

      $$\frac{\left(x^{2} + 1\right)^{6}}{6}$$

   Method #2

   1. Rewrite the integrand:

      $$2 x \left(x^{2} + 1\right)^{5} = 2 x^{11} + 10 x^{9} + 20 x^{7} + 20 x^{5} + 10 x^{3} + 2 x$$

   2. Integrate term-by-term:

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

         $$\int 2 x^{11}\, dx = 2 \int x^{11}\, dx$$

         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: $\frac{x^{12}}{6}$

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

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

         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: $x^{10}$

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

         $$\int 20 x^{7}\, dx = 20 \int x^{7}\, dx$$

         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: $\frac{5 x^{8}}{2}$

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

         $$\int 20 x^{5}\, dx = 20 \int x^{5}\, dx$$

         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: $\frac{10 x^{6}}{3}$

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

         $$\int 10 x^{3}\, dx = 10 \int x^{3}\, dx$$

         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: $\frac{5 x^{4}}{2}$

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

         $$\int 2 x\, dx = 2 \int x\, dx$$

         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: $x^{2}$

      The result is: $\frac{x^{12}}{6} + x^{10} + \frac{5 x^{8}}{2} + \frac{10 x^{6}}{3} + \frac{5 x^{4}}{2} + x^{2}$

2. Now simplify:

   $$\frac{\left(x^{2} + 1\right)^{6}}{6}$$

3. Add the constant of integration:

   $$\frac{\left(x^{2} + 1\right)^{6}}{6}+ \mathrm{constant}$$

The answer is:

$$\frac{\left(x^{2} + 1\right)^{6}}{6}+ \mathrm{constant}$$