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 1 dx + 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 $