Question

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