Question

Evaluate: $\int \frac{3x^2+1}{(x^2-1{)}^3}dx$

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

Answer 1

The correct option is A $C-\frac{x}{{(x^2-1)}^2}$

Given, $\int \frac{3x^2+1}{{(x^2-1)}^3}dx=\int \frac{3x^2+1}{{(x-1)}^3{(x+1)}^3}dx$

$=-\frac{1}{2}\int (\frac{1}{{(x+1)}^3}-\frac{1}{{(x-1)}^3})dx$

$=\frac{1}{2}\{\frac{{(x-1)}^{-2}}{-1}+\frac{{(x+1)}^{-2}}{-2}\}+C$

$=C-\frac{x}{{(x^2-1)}^2}$