Integral of (4x-3)(2x-5)dx dx

1. There are multiple ways to do this integral.

   Method #1

   1. Let $u = 2 x$.

      Then let $du = 2 dx$ and substitute $du$:

      $$\int \left(u^{2} - \frac{13 u}{2} + \frac{15}{2}\right)\, du$$

      1. Integrate term-by-term:

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

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

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

            $$\int \left(- \frac{13 u}{2}\right)\, du = - \frac{13 \int u\, du}{2}$$

            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: $- \frac{13 u^{2}}{4}$

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

            $$\int \frac{15}{2}\, du = \frac{15 u}{2}$$

         The result is: $\frac{u^{3}}{3} - \frac{13 u^{2}}{4} + \frac{15 u}{2}$

      Now substitute $u$ back in:

      $$\frac{8 x^{3}}{3} - 13 x^{2} + 15 x$$

   Method #2

   1. Rewrite the integrand:

      $$\left(2 x - 5\right) \left(4 x - 3\right) = 8 x^{2} - 26 x + 15$$

   2. Integrate term-by-term:

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

         $$\int 8 x^{2}\, dx = 8 \int x^{2}\, dx$$

         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: $\frac{8 x^{3}}{3}$

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

         $$\int \left(- 26 x\right)\, dx = - 26 \int x\, dx$$

         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: $- 13 x^{2}$

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

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

      The result is: $\frac{8 x^{3}}{3} - 13 x^{2} + 15 x$

2. Now simplify:

   $$\frac{x \left(8 x^{2} - 39 x + 45\right)}{3}$$

3. Add the constant of integration:

   $$\frac{x \left(8 x^{2} - 39 x + 45\right)}{3}+ \mathrm{constant}$$

The answer is:

$$\frac{x \left(8 x^{2} - 39 x + 45\right)}{3}+ \mathrm{constant}$$