Question

$\int \frac{x\ln x}{{(x^2-1)}^{3/2}}dx$ equals

• A. $arc\sec x-\frac{\ln x}{\sqrt{x^2-1}}+C$
• B. ${\sec }^{-1}x+\frac{\ln x}{\sqrt{x^2-1}}+C$
• C. ${\cos }^{-1}x-\frac{\ln x}{\sqrt{x^2-1}}+C$
• D. $\sec x-\frac{\ln x}{\sqrt{x^2-1}}+C$

Answer 1

Answer: (B)

${\sec }^{-1}x+\frac{\ln x}{\sqrt{x^2-1}}+C$

We have

$\int \frac{x\ln x}{{\left(x^2-\right)}^{\frac{3}{2}}}dx$

$=\frac{1}{2}\left[\frac{2x\ln x}{{\left(x^2-1\right)}^{\frac{3}{2}}}+2\int \frac{1}{x{\left(x^2-1\right)}^{\frac{1}{2}}}dx\right]$

$={\sec }^{-1}x+\frac{\ln x}{\sqrt{x^2-1}}+c$

Hence, the $\left(B\right)$ is the correct answer.