Integral of (x-1)^5 dx

1. There are multiple ways to do this integral.

   Method #1

   1. Let $u = x - 1$.

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

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

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

         $$\int u^{5}\, du = \frac{u^{6}}{6}$$

      Now substitute $u$ back in:

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

   Method #2

   1. Rewrite the integrand:

      $$\left(x - 1\right)^{5} = x^{5} - 5 x^{4} + 10 x^{3} - 10 x^{2} + 5 x - 1$$

   2. Integrate term-by-term:

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

         $$\int x^{5}\, dx = \frac{x^{6}}{6}$$

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

         $$\int \left(- 5 x^{4}\right)\, dx = - 5 \int x^{4}\, dx$$

         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: $- x^{5}$

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

         $$\int 10 x^{3}\, dx = 10 \int x^{3}\, dx$$

         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: $\frac{5 x^{4}}{2}$

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

         $$\int \left(- 10 x^{2}\right)\, dx = - 10 \int x^{2}\, dx$$

         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: $- \frac{10 x^{3}}{3}$

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

         $$\int 5 x\, dx = 5 \int x\, dx$$

         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: $\frac{5 x^{2}}{2}$

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

         $$\int \left(-1\right)\, dx = - x$$

      The result is: $\frac{x^{6}}{6} - x^{5} + \frac{5 x^{4}}{2} - \frac{10 x^{3}}{3} + \frac{5 x^{2}}{2} - x$

2. Now simplify:

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

3. Add the constant of integration:

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

The answer is: