Integral of (x³+2x)^5(6x²+4) dx

1. There are multiple ways to do this integral.

   Method #1

   1. Let $u = x^{3} + 2 x$.

      Then let $du = \left(3 x^{2} + 2\right) dx$ and substitute $2 du$:

      $$\int 4 u^{5}\, du$$

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

         $$\int 2 u^{5}\, du = 2 \int u^{5}\, du$$

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

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

         So, the result is: $\frac{u^{6}}{3}$

      Now substitute $u$ back in:

      $$\frac{\left(x^{3} + 2 x\right)^{6}}{3}$$

   Method #2

   1. Rewrite the integrand:

      $$\left(6 x^{2} + 4\right) \left(x^{3} + 2 x\right)^{5} = 6 x^{17} + 64 x^{15} + 280 x^{13} + 640 x^{11} + 800 x^{9} + 512 x^{7} + 128 x^{5}$$

   2. Integrate term-by-term:

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

         $$\int 6 x^{17}\, dx = 6 \int x^{17}\, dx$$

         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: $\frac{x^{18}}{3}$

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

         $$\int 64 x^{15}\, dx = 64 \int x^{15}\, dx$$

         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: $4 x^{16}$

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

         $$\int 280 x^{13}\, dx = 280 \int x^{13}\, dx$$

         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: $20 x^{14}$

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

         $$\int 640 x^{11}\, dx = 640 \int x^{11}\, dx$$

         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: $\frac{160 x^{12}}{3}$

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

         $$\int 800 x^{9}\, dx = 800 \int x^{9}\, dx$$

         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: $80 x^{10}$

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

         $$\int 512 x^{7}\, dx = 512 \int x^{7}\, dx$$

         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: $64 x^{8}$

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

         $$\int 128 x^{5}\, dx = 128 \int x^{5}\, dx$$

         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: $\frac{64 x^{6}}{3}$

      The result is: $\frac{x^{18}}{3} + 4 x^{16} + 20 x^{14} + \frac{160 x^{12}}{3} + 80 x^{10} + 64 x^{8} + \frac{64 x^{6}}{3}$

2. Now simplify:

   $$\frac{x^{6} \left(x^{2} + 2\right)^{6}}{3}$$

3. Add the constant of integration:

   $$\frac{x^{6} \left(x^{2} + 2\right)^{6}}{3}+ \mathrm{constant}$$

The answer is:

$$\frac{x^{6} \left(x^{2} + 2\right)^{6}}{3}+ \mathrm{constant}$$