Integral of (3x+5)^3 dx

1. There are multiple ways to do this integral.

   Method #1

   1. Let $u = 3 x + 5$.

      Then let $du = 3 dx$ and substitute $\frac{du}{3}$:

      $$\int \frac{u^{3}}{9}\, du$$

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

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

         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: $\frac{u^{4}}{12}$

      Now substitute $u$ back in:

      $$\frac{\left(3 x + 5\right)^{4}}{12}$$

   Method #2

   1. Rewrite the integrand:

      $$\left(3 x + 5\right)^{3} = 27 x^{3} + 135 x^{2} + 225 x + 125$$

   2. Integrate term-by-term:

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

         $$\int 27 x^{3}\, dx = 27 \int x^{3}\, dx$$

         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: $\frac{27 x^{4}}{4}$

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

         $$\int 135 x^{2}\, dx = 135 \int x^{2}\, dx$$

         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: $45 x^{3}$

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

         $$\int 225 x\, dx = 225 \int x\, dx$$

         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: $\frac{225 x^{2}}{2}$

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

         $$\int 125\, dx = 125 x$$

      The result is: $\frac{27 x^{4}}{4} + 45 x^{3} + \frac{225 x^{2}}{2} + 125 x$

2. Now simplify:

   $$\frac{\left(3 x + 5\right)^{4}}{12}$$

3. Add the constant of integration:

   $$\frac{\left(3 x + 5\right)^{4}}{12}+ \mathrm{constant}$$

The answer is:

$$\frac{\left(3 x + 5\right)^{4}}{12}+ \mathrm{constant}$$