Question

If $I=\displaystyle \int \frac{d x}{\sqrt[3]{x^{\frac{5}{2}} \left(1 + x\right)^{\frac{7}{2}}}}$ $=kf\left(x\right)+c,$ where $c$ is the integration constant and $f\left(1\right)=\frac{1}{2^{\frac{1}{6}}},$ then the value of $f\left(2\right)$ is equal to

• A. $6\left(\frac{2}{3}\right)^{\frac{1}{6}}$
• B. $6 \left(\frac{3}{2}\right)^{\frac{1}{6}}$
• C. $\left(\frac{2}{3}\right)^{\frac{1}{6}}$
• D. $\left(\frac{2}{3}\right)^{6}$

Answer 1

Answer: (C)

The given integral is $I=\displaystyle \int \frac{d x}{\left(\left[x^{6} \left(\frac{1 + x}{x}\right)^{\frac{7}{2}}\right]\right)^{\frac{1}{3}}}$
$\Rightarrow I=\displaystyle \int \frac{d x}{x^{2} \left(1 + \frac{1}{x}\right)^{\frac{7}{6}}}$
Let, $1+\frac{1}{x}=t$
$\Rightarrow -\frac{d x}{x^{2}}=dt$
$\Rightarrow I=\displaystyle \int - \frac{d t}{t^{\frac{7}{6}}}$
$=6t^{- \frac{1}{6}}+c$
$=6\left(\frac{x}{x + 1}\right)^{\frac{1}{6}}+c$
$\therefore f\left(x\right)=\left(\frac{x}{x + 1}\right)^{\frac{1}{6}}$
$\Rightarrow f\left(2\right)=\left(\frac{2}{3}\right)^{\frac{1}{6}}$