Integral of (6-7x)⁵dx dx

1. There are multiple ways to do this integral.

   Method #1

   1. Let $u = 6 - 7 x$.

      Then let $du = - 7 dx$ and substitute $- \frac{du}{7}$:

      $$\int \left(- \frac{u^{5}}{7}\right)\, du$$

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

         $$\int u^{5}\, du = - \frac{\int u^{5}\, du}{7}$$

         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: $- \frac{u^{6}}{42}$

      Now substitute $u$ back in:

      $$- \frac{\left(6 - 7 x\right)^{6}}{42}$$

   Method #2

   1. Rewrite the integrand:

      $$\left(6 - 7 x\right)^{5} = - 16807 x^{5} + 72030 x^{4} - 123480 x^{3} + 105840 x^{2} - 45360 x + 7776$$

   2. Integrate term-by-term:

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

         $$\int \left(- 16807 x^{5}\right)\, dx = - 16807 \int x^{5}\, dx$$

         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: $- \frac{16807 x^{6}}{6}$

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

         $$\int 72030 x^{4}\, dx = 72030 \int x^{4}\, dx$$

         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: $14406 x^{5}$

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

         $$\int \left(- 123480 x^{3}\right)\, dx = - 123480 \int x^{3}\, dx$$

         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: $- 30870 x^{4}$

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

         $$\int 105840 x^{2}\, dx = 105840 \int x^{2}\, dx$$

         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: $35280 x^{3}$

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

         $$\int \left(- 45360 x\right)\, dx = - 45360 \int x\, dx$$

         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: $- 22680 x^{2}$

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

         $$\int 7776\, dx = 7776 x$$

      The result is: $- \frac{16807 x^{6}}{6} + 14406 x^{5} - 30870 x^{4} + 35280 x^{3} - 22680 x^{2} + 7776 x$

2. Now simplify:

   $$- \frac{\left(7 x - 6\right)^{6}}{42}$$

3. Add the constant of integration:

   $$- \frac{\left(7 x - 6\right)^{6}}{42}+ \mathrm{constant}$$

The answer is:

$$- \frac{\left(7 x - 6\right)^{6}}{42}+ \mathrm{constant}$$