Question

Integrate:
$\frac{x^3+x+1}{x^2-1}$

Answer 1

$\begin{array}{c}\int \frac{x^3=x+1}{x^2-1}dx\\ \Rightarrow \int \frac{2x+1}{\left(x^2-1\right)}+\int xdx\\ \Rightarrow 2\int \frac{xdx}{\left(x^2-1\right)}+\int \frac{dx}{x^2-1^2}+\int xdx\\ let\left(x^2-1\right)=t\\ 2xdx=dt\\ dx=\frac{dt}{2x}\\ \Rightarrow \int \frac{dt}{t}+\frac{1}{2}\log \left|\frac{x-1}{x+1}\right|+\frac{x^2}{2}+C\\ \Rightarrow \log t+\frac{1}{2}\log \left|\frac{x-1}{x+1}\right|+\frac{x^2}{2}+C\\ \Rightarrow \log \left(x^2-1\right)+\frac{1}{2}\log \left|\frac{x-1}{x+1}\right|+\frac{x^2}{2}+C\end{array}$