Integral of (x^4+3)*(4x^3dx) dx

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

   $$\int \left(x^{4} + 3\right) 4 x^{3} \cdot 1\, dx = 4 \int x^{3} \left(x^{4} + 3\right)\, dx$$

   1. There are multiple ways to do this integral.

      Method #1

      1. Let $u = x^{4}$.

         Then let $du = 4 x^{3} dx$ and substitute $du$:

         $$\int \left(\frac{u}{4} + \frac{3}{4}\right)\, du$$

         1. Integrate term-by-term:

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

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

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

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

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

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

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

            The result is: $\frac{u^{2}}{8} + \frac{3 u}{4}$

         Now substitute $u$ back in:

         $$\frac{x^{8}}{8} + \frac{3 x^{4}}{4}$$

      Method #2

      1. Rewrite the integrand:

         $$x^{3} \left(x^{4} + 3\right) = x^{7} + 3 x^{3}$$

      2. Integrate term-by-term:

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

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

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

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

            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: $\frac{3 x^{4}}{4}$

         The result is: $\frac{x^{8}}{8} + \frac{3 x^{4}}{4}$

   So, the result is: $\frac{x^{8}}{2} + 3 x^{4}$

2. Now simplify:

   $$\frac{x^{4} \left(x^{4} + 6\right)}{2}$$

3. Add the constant of integration:

   $$\frac{x^{4} \left(x^{4} + 6\right)}{2}+ \mathrm{constant}$$

The answer is:

$$\frac{x^{4} \left(x^{4} + 6\right)}{2}+ \mathrm{constant}$$