Question

The value of $\int \frac{x^2+1}{x^2-1}dx$ is

• A. $Log\left(\frac{x+1}{x-1}\right)+C$
• B. $Log\left(\frac{x-1}{x+1}\right)+C$
• C. $Log(x^2-1)+C$
• D. $x+Log\left(\frac{x-1}{x+1}\right)+C$

Answer 1

Answer: (D)

Let $I=\int \frac{x^2+1}{x^2-1}dx$
$\Rightarrow I=\int \frac{x^2+1-1+1}{x^2-1}dx$
$\Rightarrow I=\int \frac{x^2-1}{x^2-1}dx+\int \frac{2}{x^2-1}dx$
$\Rightarrow I=\int 1dx+2\int \frac{1}{x^2-1}dx$
$\Rightarrow I=x+2\frac{1}{2}\log \left(\frac{x-1}{x+1}\right)+c$
$\Rightarrow I=x+\log \left(\frac{x-1}{x+1}\right)+c$