Integral of (3x+1)^5 dx

1. There are multiple ways to do this integral.

   Method #1

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

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

      $$\int \frac{u^{5}}{9}\, du$$

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

         $$\int \frac{u^{5}}{3}\, du = \frac{\int u^{5}\, du}{3}$$

         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: $\frac{u^{6}}{18}$

      Now substitute $u$ back in:

      $$\frac{\left(3 x + 1\right)^{6}}{18}$$

   Method #2

   1. Rewrite the integrand:

      $$\left(3 x + 1\right)^{5} = 243 x^{5} + 405 x^{4} + 270 x^{3} + 90 x^{2} + 15 x + 1$$

   2. Integrate term-by-term:

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

         $$\int 243 x^{5}\, dx = 243 \int x^{5}\, dx$$

         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: $\frac{81 x^{6}}{2}$

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

         $$\int 405 x^{4}\, dx = 405 \int x^{4}\, dx$$

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

            $$\int x^{4}\, dx = \frac{x^{5}}{5}$$

         So, the result is: $81 x^{5}$

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

         $$\int 270 x^{3}\, dx = 270 \int x^{3}\, dx$$

         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: $\frac{135 x^{4}}{2}$

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

         $$\int 90 x^{2}\, dx = 90 \int x^{2}\, dx$$

         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: $30 x^{3}$

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

         $$\int 15 x\, dx = 15 \int x\, dx$$

         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: $\frac{15 x^{2}}{2}$

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

         $$\int 1\, dx = x$$

      The result is: $\frac{81 x^{6}}{2} + 81 x^{5} + \frac{135 x^{4}}{2} + 30 x^{3} + \frac{15 x^{2}}{2} + x$

2. Now simplify:

   $$\frac{\left(3 x + 1\right)^{6}}{18}$$

3. Add the constant of integration:

   $$\frac{\left(3 x + 1\right)^{6}}{18}+ \mathrm{constant}$$

The answer is:

$$\frac{\left(3 x + 1\right)^{6}}{18}+ \mathrm{constant}$$