Integral of ax(1-x) dx

1. There are multiple ways to do this integral.

   Method #1

   1. Let $u = - x$.

      Then let $du = - dx$ and substitute $du$:

      $$\int \left(a u^{2} + a u\right)\, du$$

      1. Integrate term-by-term:

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

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

            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: $\frac{u^{3} a}{3}$

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

            $$\int u a\, du = a \int u\, du$$

            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: $\frac{u^{2} a}{2}$

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

      Now substitute $u$ back in:

      $$- \frac{a x^{3}}{3} + \frac{a x^{2}}{2}$$

   Method #2

   1. Rewrite the integrand:

      $$a x \left(1 - x\right) = - a x^{2} + a x$$

   2. Integrate term-by-term:

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

         $$\int \left(- a x^{2}\right)\, dx = - a \int x^{2}\, dx$$

         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: $- \frac{a x^{3}}{3}$

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

         $$\int a x\, dx = a \int x\, dx$$

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

      The result is: $- \frac{a x^{3}}{3} + \frac{a x^{2}}{2}$

2. Now simplify:

   $$\frac{a x^{2} \left(3 - 2 x\right)}{6}$$

3. Add the constant of integration:

   $$\frac{a x^{2} \left(3 - 2 x\right)}{6}+ \mathrm{constant}$$

The answer is:

$$\frac{a x^{2} \left(3 - 2 x\right)}{6}+ \mathrm{constant}$$