Question

If the value of integral $\displaystyle \int \frac{d x}{\left(x + \sqrt{x^{2} - 1}\right)^{2}}=ax^{3}-x+b\left(x^{2} - 1\right)^{\frac{1}{b}}+C$ , (where, $C$ is the constant of integration), then $a\times b$ is equal to

• A. $1$
• B. $\frac{4}{9}$
• C. $2$
• D. $\frac{9}{4}$

Answer 1

Answer: (B)

$\displaystyle \int \left(x - \sqrt{x^{2} - 1}\right)^{2} d x=\displaystyle \int x^{2} + \left(x^{2} - 1\right) - 2 x \sqrt{x^{2} - 1} d x$
$=\frac{2 x^{3}}{3}-x+\frac{2}{3}\left(x^{2} - 1\right)^{\frac{3}{2}}+C$ $\Rightarrow a=b=\frac{2}{3}$