Resposta:

Primeiramente divida o numerador pelo denominador. Suponho que você saiba.

[tex]\displaystyle\int (\frac{x^2+x-1}{x^2-1})dx= \displaystyle\int (1 +\frac{x}{x^2-1})dx\\\\x^2-1=u\\\displaystyle\int 2xdx=du \implies xdx=\frac{1}{2}du\\\\ \displaystyle\int (1+\frac{x}{x^2-1})dx= \displaystyle\int dx+\displaystyle\int \frac{xdx}{x^2-1}=x+\displaystyle\int \frac{\frac{1}{2}du }{u} =x+\frac{1}{2} \displaystyle\int \frac{du}{u} =\\\\x+\frac{1}{2} ln|u|=c=x+\frac{1}{2} ln|x^2-1|+c[/tex]