Integral of /(3x+1)^2-5x dx

1. Integrate term-by-term:

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

      $$\int \left(- 5 x\right)\, dx = - \int 5 x\, dx$$

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

         $$\int 5 x\, dx = 5 \int x\, dx$$

         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: $\frac{5 x^{2}}{2}$

      So, the result is: $- \frac{5 x^{2}}{2}$

   1. There are multiple ways to do this integral.

      Method #1

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

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

         $$\int \frac{u^{2}}{9}\, du$$

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

            $$\int \frac{u^{2}}{3}\, du = \frac{\int u^{2}\, du}{3}$$

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

               $$\int u^{2}\, du = \frac{u^{3}}{3}$$

            So, the result is: $\frac{u^{3}}{9}$

         Now substitute $u$ back in:

         $$\frac{\left(3 x + 1\right)^{3}}{9}$$

      Method #2

      1. Rewrite the integrand:

         $$\left(3 x + 1\right)^{2} = 9 x^{2} + 6 x + 1$$

      2. Integrate term-by-term:

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

            $$\int 9 x^{2}\, dx = 9 \int x^{2}\, dx$$

            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: $3 x^{3}$

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

            $$\int 6 x\, dx = 6 \int x\, dx$$

            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: $3 x^{2}$

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

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

         The result is: $3 x^{3} + 3 x^{2} + x$

   The result is: $- \frac{5 x^{2}}{2} + \frac{\left(3 x + 1\right)^{3}}{9}$

2. Now simplify:

   $$- \frac{5 x^{2}}{2} + \frac{\left(3 x + 1\right)^{3}}{9}$$

3. Add the constant of integration:

   $$- \frac{5 x^{2}}{2} + \frac{\left(3 x + 1\right)^{3}}{9}+ \mathrm{constant}$$

The answer is:

$$- \frac{5 x^{2}}{2} + \frac{\left(3 x + 1\right)^{3}}{9}+ \mathrm{constant}$$