Question

$\int \frac{x^2+1}{x^2-1}dx=\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_.$

Answer 1

$\int \frac{x^2+1}{x^2-1}dx$

$=\int \frac{x^2-1+2}{x^2-1}dx$

$=\int \frac{x^2-1}{x^2-1}dx+\int \frac{2}{x^2-1}dx$

$=\int dx+2\int \frac{1}{x^2-1}dx$

$=x+2\times \frac{1}{2}\log \left|\frac{x-1}{x+1}\right|+C\left(\int \frac{1}{x^2-a^2}dx=\frac{1}{2a}\log \left|\frac{x-a}{x+a}\right|+C\right)$

$=x+\log \left|\frac{x-1}{x+1}\right|+C$


$\int \frac{x^2+1}{x^2-1}dx=\underline{x+\log \left|\frac{x-1}{x+1}\right|+C}.$