Question

If the integral $I=\int \frac{x\sqrt{x}-3x+3\sqrt{x}-1}{x-2\sqrt{x}+1}dx$ $=f\left(x\right)+C$ (where, $x>0$ and $C$ is the constant of integration) and $f\left(1\right)=\frac{-1}{3}$ , then the value of $f\left(9\right)$ is equal to

• A. 3
• B. 6
• C. 9
• D. 12

Answer 1

Answer: (C)

Given integral is $I=\int \frac{{\left(\sqrt{x}-1\right)}^3}{{\left(\sqrt{x}-1\right)}^2}dx$
$=\int \left(\sqrt{x}-1\right)dx$
$=\frac{2}{3}{x}^{\frac{3}{2}}-x+C$
$\therefore f\left(x\right)=\frac{2}{3}{x}^{\frac{3}{2}}-x$
$\Rightarrow f\left(9\right)=18-9=9$