16) $$ \int \frac{x^{5}+1}{x-1} d x $$

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Calculate the integral $$ \int \frac{x^{5}+1}{x-1} dx $$.
Evaluate the integral by following steps:
- step0: Evaluate using formulas and rules:
  $$\int \frac{x^{5}+1}{x-1} dx$$
- step1: Rewrite the fraction:
  $$\int x^{4}+x^{3}+x^{2}+x+1+\frac{2}{x-1} dx$$
- step2: Use properties of integrals:
  $$\int x^{4}+x^{3}+x^{2}+x+1 dx+\int \frac{2}{x-1} dx$$
- step3: Evaluate the integral:
  $$\frac{x^{5}}{5}+\frac{x^{4}}{4}+\frac{x^{3}}{3}+\frac{x^{2}}{2}+x+\int \frac{2}{x-1} dx$$
- step4: Evaluate the integral:
  $$\frac{x^{5}}{5}+\frac{x^{4}}{4}+\frac{x^{3}}{3}+\frac{x^{2}}{2}+x+2\ln{\left(\left|x-1\right|\right)}$$
- step5: Add the constant of integral C:
  $$\frac{x^{5}}{5}+\frac{x^{4}}{4}+\frac{x^{3}}{3}+\frac{x^{2}}{2}+x+2\ln{\left(\left|x-1\right|\right)} + C, C \in \mathbb{R}$$
The integral of $$ \frac{x^{5}+1}{x-1} $$ with respect to $$ x $$ is $$ \frac{x^{5}}{5} + \frac{x^{4}}{4} + \frac{x^{3}}{3} + \frac{x^{2}}{2} + x + 2\ln{|x-1|} + C $$, where $$ C $$ is an arbitrary constant.
The integral of $$ \frac{x^{5}+1}{x-1} $$ with respect to $$ x $$ is $$ \frac{x^{5}}{5} + \frac{x^{4}}{4} + \frac{x^{3}}{3} + \frac{x^{2}}{2} + x + 2\ln{|x-1|} + C $$, where $$ C $$ is a constant.

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