Question

If $I=\displaystyle \int \frac{\left(ln x\right)^{5}}{\sqrt{x^{2} + x^{2} \left(ln ⁡ x\right)^{3}}}dx=$ $k\sqrt{\left(ln x\right)^{3} + 1}\left(\left(ln ⁡ x\right)^{3} - 2\right)+c$ (where $c$ is the constant of integration), then $9k$ is equal to

• A. $4$
• B. $2$
• C. $6$
• D. $10$

Answer 1

Answer: (B)

Let $1+\left(ln x\right)^{3}=t^{2}$
$\Rightarrow I=\displaystyle \int \frac{\left(t^{2} - 1\right) 2 t d t}{3 t}$
$\Rightarrow I=\frac{2}{3}\displaystyle \int \left(t^{2} - 1\right)dt$
$=\frac{2 t^{3}}{9}-\frac{2}{3}t+c$
$=\frac{2}{9}t\left(t^{2} - 3\right)+c$
$=\frac{2}{9}\sqrt{1 + \left(ln x\right)^{3}}\left(\left(ln ⁡ x\right)^{3} - 2\right)+c$
$\therefore k=\frac{2}{9}\Rightarrow 9k=2$