Integral of (0.291-x*0.169^2)*x^1 dx

1. There are multiple ways to do this integral.

   Method #1

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

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

      $$\int \left(- \frac{28561 u^{2}}{1000000} + \frac{291 u}{1000}\right)\, du$$

      1. Integrate term-by-term:

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

            $$\int \left(- \frac{28561 u^{2}}{1000000}\right)\, du = - \frac{28561 \int u^{2}\, du}{1000000}$$

            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: $- \frac{28561 u^{3}}{3000000}$

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

            $$\int \frac{291 u}{1000}\, du = \frac{291 \int u\, du}{1000}$$

            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: $\frac{291 u^{2}}{2000}$

         The result is: $- \frac{28561 u^{3}}{3000000} + \frac{291 u^{2}}{2000}$

      Now substitute $u$ back in:

      $$- \frac{28561 x^{3}}{3000000} + \frac{291 x^{2}}{2000}$$

   Method #2

   1. Rewrite the integrand:

      $$x^{1} \left(- \left(\frac{169}{1000}\right)^{2} x + \frac{291}{1000}\right) = - \frac{28561 x^{2}}{1000000} + \frac{291 x}{1000}$$

   2. Integrate term-by-term:

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

         $$\int \left(- \frac{28561 x^{2}}{1000000}\right)\, dx = - \frac{28561 \int x^{2}\, dx}{1000000}$$

         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: $- \frac{28561 x^{3}}{3000000}$

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

         $$\int \frac{291 x}{1000}\, dx = \frac{291 \int x\, dx}{1000}$$

         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: $\frac{291 x^{2}}{2000}$

      The result is: $- \frac{28561 x^{3}}{3000000} + \frac{291 x^{2}}{2000}$

2. Now simplify:

   $$\frac{x^{2} \left(436500 - 28561 x\right)}{3000000}$$

3. Add the constant of integration:

   $$\frac{x^{2} \left(436500 - 28561 x\right)}{3000000}+ \mathrm{constant}$$

The answer is:

$$\frac{x^{2} \left(436500 - 28561 x\right)}{3000000}+ \mathrm{constant}$$