Integral of 6x²(2x³+3)³dx dx

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

   $$\int 6 x^{2} \left(2 x^{3} + 3\right)^{3} \cdot 1\, dx = 6 \int x^{2} \left(2 x^{3} + 3\right)^{3}\, dx$$

   1. There are multiple ways to do this integral.

      Method #1

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

         Then let $du = 6 x^{2} dx$ and substitute $\frac{du}{6}$:

         $$\int \frac{u^{3}}{36}\, du$$

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

            $$\int \frac{u^{3}}{6}\, du = \frac{\int u^{3}\, du}{6}$$

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

               $$\int u^{3}\, du = \frac{u^{4}}{4}$$

            So, the result is: $\frac{u^{4}}{24}$

         Now substitute $u$ back in:

         $$\frac{\left(2 x^{3} + 3\right)^{4}}{24}$$

      Method #2

      1. Rewrite the integrand:

         $$x^{2} \left(2 x^{3} + 3\right)^{3} = 8 x^{11} + 36 x^{8} + 54 x^{5} + 27 x^{2}$$

      2. Integrate term-by-term:

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

            $$\int 8 x^{11}\, dx = 8 \int x^{11}\, dx$$

            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: $\frac{2 x^{12}}{3}$

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

            $$\int 36 x^{8}\, dx = 36 \int x^{8}\, dx$$

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

               $$\int x^{8}\, dx = \frac{x^{9}}{9}$$

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

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

            $$\int 54 x^{5}\, dx = 54 \int x^{5}\, dx$$

            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: $9 x^{6}$

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

            $$\int 27 x^{2}\, dx = 27 \int x^{2}\, dx$$

            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: $9 x^{3}$

         The result is: $\frac{2 x^{12}}{3} + 4 x^{9} + 9 x^{6} + 9 x^{3}$

   So, the result is: $\frac{\left(2 x^{3} + 3\right)^{4}}{4}$

2. Now simplify:

   $$\frac{\left(2 x^{3} + 3\right)^{4}}{4}$$

3. Add the constant of integration:

   $$\frac{\left(2 x^{3} + 3\right)^{4}}{4}+ \mathrm{constant}$$

The answer is:

$$\frac{\left(2 x^{3} + 3\right)^{4}}{4}+ \mathrm{constant}$$