Integral of 5(5x-10)^5 dx

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

   $$\int 5 \left(5 x - 10\right)^{5}\, dx = 5 \int \left(5 x - 10\right)^{5}\, dx$$

   1. There are multiple ways to do this integral.

      Method #1

      1. Let $u = 5 x - 10$.

         Then let $du = 5 dx$ and substitute $\frac{du}{5}$:

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

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

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

            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: $\frac{u^{6}}{30}$

         Now substitute $u$ back in:

         $$\frac{\left(5 x - 10\right)^{6}}{30}$$

      Method #2

      1. Rewrite the integrand:

         $$\left(5 x - 10\right)^{5} = 3125 x^{5} - 31250 x^{4} + 125000 x^{3} - 250000 x^{2} + 250000 x - 100000$$

      2. Integrate term-by-term:

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

            $$\int 3125 x^{5}\, dx = 3125 \int x^{5}\, dx$$

            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: $\frac{3125 x^{6}}{6}$

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

            $$\int \left(- 31250 x^{4}\right)\, dx = - 31250 \int x^{4}\, dx$$

            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: $- 6250 x^{5}$

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

            $$\int 125000 x^{3}\, dx = 125000 \int x^{3}\, dx$$

            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: $31250 x^{4}$

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

            $$\int \left(- 250000 x^{2}\right)\, dx = - 250000 \int x^{2}\, dx$$

            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: $- \frac{250000 x^{3}}{3}$

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

            $$\int 250000 x\, dx = 250000 \int x\, dx$$

            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: $125000 x^{2}$

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

            $$\int \left(-100000\right)\, dx = - 100000 x$$

         The result is: $\frac{3125 x^{6}}{6} - 6250 x^{5} + 31250 x^{4} - \frac{250000 x^{3}}{3} + 125000 x^{2} - 100000 x$

   So, the result is: $\frac{\left(5 x - 10\right)^{6}}{6}$

2. Now simplify:

   $$\frac{15625 \left(x - 2\right)^{6}}{6}$$

3. Add the constant of integration:

   $$\frac{15625 \left(x - 2\right)^{6}}{6}+ \mathrm{constant}$$

The answer is:

$$\frac{15625 \left(x - 2\right)^{6}}{6}+ \mathrm{constant}$$