Integral of x^5(x^6-1)^9 dx

1. There are multiple ways to do this integral.

   Method #1

   1. Let $u = x^{6} - 1$.

      Then let $du = 6 x^{5} dx$ and substitute $\frac{du}{6}$:

      $$\int \frac{u^{9}}{6}\, du$$

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

         $$\int u^{9}\, du = \frac{\int u^{9}\, du}{6}$$

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

            $$\int u^{9}\, du = \frac{u^{10}}{10}$$

         So, the result is: $\frac{u^{10}}{60}$

      Now substitute $u$ back in:

      $$\frac{\left(x^{6} - 1\right)^{10}}{60}$$

   Method #2

   1. Rewrite the integrand:

      $$x^{5} \left(x^{6} - 1\right)^{9} = x^{59} - 9 x^{53} + 36 x^{47} - 84 x^{41} + 126 x^{35} - 126 x^{29} + 84 x^{23} - 36 x^{17} + 9 x^{11} - x^{5}$$

   2. Integrate term-by-term:

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

         $$\int x^{59}\, dx = \frac{x^{60}}{60}$$

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

         $$\int \left(- 9 x^{53}\right)\, dx = - 9 \int x^{53}\, dx$$

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

            $$\int x^{53}\, dx = \frac{x^{54}}{54}$$

         So, the result is: $- \frac{x^{54}}{6}$

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

         $$\int 36 x^{47}\, dx = 36 \int x^{47}\, dx$$

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

            $$\int x^{47}\, dx = \frac{x^{48}}{48}$$

         So, the result is: $\frac{3 x^{48}}{4}$

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

         $$\int \left(- 84 x^{41}\right)\, dx = - 84 \int x^{41}\, dx$$

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

            $$\int x^{41}\, dx = \frac{x^{42}}{42}$$

         So, the result is: $- 2 x^{42}$

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

         $$\int 126 x^{35}\, dx = 126 \int x^{35}\, dx$$

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

            $$\int x^{35}\, dx = \frac{x^{36}}{36}$$

         So, the result is: $\frac{7 x^{36}}{2}$

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

         $$\int \left(- 126 x^{29}\right)\, dx = - 126 \int x^{29}\, dx$$

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

            $$\int x^{29}\, dx = \frac{x^{30}}{30}$$

         So, the result is: $- \frac{21 x^{30}}{5}$

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

         $$\int 84 x^{23}\, dx = 84 \int x^{23}\, dx$$

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

            $$\int x^{23}\, dx = \frac{x^{24}}{24}$$

         So, the result is: $\frac{7 x^{24}}{2}$

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

         $$\int \left(- 36 x^{17}\right)\, dx = - 36 \int x^{17}\, dx$$

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

            $$\int x^{17}\, dx = \frac{x^{18}}{18}$$

         So, the result is: $- 2 x^{18}$

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

         $$\int 9 x^{11}\, dx = 9 \int x^{11}\, dx$$

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

            $$\int x^{11}\, dx = \frac{x^{12}}{12}$$

         So, the result is: $\frac{3 x^{12}}{4}$

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

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

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

      The result is: $\frac{x^{60}}{60} - \frac{x^{54}}{6} + \frac{3 x^{48}}{4} - 2 x^{42} + \frac{7 x^{36}}{2} - \frac{21 x^{30}}{5} + \frac{7 x^{24}}{2} - 2 x^{18} + \frac{3 x^{12}}{4} - \frac{x^{6}}{6}$

2. Now simplify:

   $$\frac{\left(x^{6} - 1\right)^{10}}{60}$$

3. Add the constant of integration:

   $$\frac{\left(x^{6} - 1\right)^{10}}{60}+ \mathrm{constant}$$

The answer is:

$$\frac{\left(x^{6} - 1\right)^{10}}{60}+ \mathrm{constant}$$