Integral of x(9-x^2)^7 dx

1. There are multiple ways to do this integral.

   Method #1

   1. Let $u = 9 - x^{2}$.

      Then let $du = - 2 x dx$ and substitute $- \frac{du}{2}$:

      $$\int \left(- \frac{u^{7}}{2}\right)\, du$$

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

         $$\int u^{7}\, du = - \frac{\int u^{7}\, du}{2}$$

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

            $$\int u^{7}\, du = \frac{u^{8}}{8}$$

         So, the result is: $- \frac{u^{8}}{16}$

      Now substitute $u$ back in:

      $$- \frac{\left(9 - x^{2}\right)^{8}}{16}$$

   Method #2

   1. Rewrite the integrand:

      $$x \left(9 - x^{2}\right)^{7} = - x^{15} + 63 x^{13} - 1701 x^{11} + 25515 x^{9} - 229635 x^{7} + 1240029 x^{5} - 3720087 x^{3} + 4782969 x$$

   2. Integrate term-by-term:

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

         $$\int \left(- x^{15}\right)\, dx = - \int x^{15}\, dx$$

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

            $$\int x^{15}\, dx = \frac{x^{16}}{16}$$

         So, the result is: $- \frac{x^{16}}{16}$

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

         $$\int 63 x^{13}\, dx = 63 \int x^{13}\, dx$$

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

            $$\int x^{13}\, dx = \frac{x^{14}}{14}$$

         So, the result is: $\frac{9 x^{14}}{2}$

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

         $$\int \left(- 1701 x^{11}\right)\, dx = - 1701 \int x^{11}\, dx$$

         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: $- \frac{567 x^{12}}{4}$

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

         $$\int 25515 x^{9}\, dx = 25515 \int x^{9}\, dx$$

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

            $$\int x^{9}\, dx = \frac{x^{10}}{10}$$

         So, the result is: $\frac{5103 x^{10}}{2}$

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

         $$\int \left(- 229635 x^{7}\right)\, dx = - 229635 \int x^{7}\, dx$$

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

            $$\int x^{7}\, dx = \frac{x^{8}}{8}$$

         So, the result is: $- \frac{229635 x^{8}}{8}$

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

         $$\int 1240029 x^{5}\, dx = 1240029 \int x^{5}\, dx$$

         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: $\frac{413343 x^{6}}{2}$

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

         $$\int \left(- 3720087 x^{3}\right)\, dx = - 3720087 \int x^{3}\, dx$$

         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: $- \frac{3720087 x^{4}}{4}$

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

         $$\int 4782969 x\, dx = 4782969 \int x\, dx$$

         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: $\frac{4782969 x^{2}}{2}$

      The result is: $- \frac{x^{16}}{16} + \frac{9 x^{14}}{2} - \frac{567 x^{12}}{4} + \frac{5103 x^{10}}{2} - \frac{229635 x^{8}}{8} + \frac{413343 x^{6}}{2} - \frac{3720087 x^{4}}{4} + \frac{4782969 x^{2}}{2}$

2. Now simplify:

   $$- \frac{\left(x^{2} - 9\right)^{8}}{16}$$

3. Add the constant of integration:

   $$- \frac{\left(x^{2} - 9\right)^{8}}{16}+ \mathrm{constant}$$

The answer is:

$$- \frac{\left(x^{2} - 9\right)^{8}}{16}+ \mathrm{constant}$$